Graph Immersions, Inverse Monoids, and Deck Transformations

TL;DR

This paper generalizes the classical correspondence between deck transformations and normalizers from graph coverings to graph immersions using inverse monoids, revealing new structural insights and extension properties.

Contribution

It extends the known theory of deck transformations from graph covers to immersions via inverse monoids, establishing an analogous normalizer relationship.

Findings

Deck transformation groups are isomorphic to the quotient of the normalizer of the submonoid by the submonoid.

A graph immersion can be extended to a cover with compatible deck transformations.

Relationship established between group actions on graphs and deck transformations.

Abstract

If $f : \tilde{\Gamma} \rightarrow \Gamma$ is a covering map between connected graphs, and $H$ is the subgroup of $\pi_1(\Gamma,v)$ used to construct the cover, then it is well known that the group of deck transformations of the cover is isomorphic to $ N(H)/H$, where $N(H)$ is the normalizer of $H$ in $\pi_1(\Gamma,v)$. We show that an entirely analogous result holds for immersions between connected graphs, where the subgroup $H$ is replaced by the closed inverse submonoid of the inverse monoid $L(\Gamma,v)$ used to construct the immersion. We observe a relationship between group actions on graphs and deck transformations of graph immersions. We also show that a graph immersion $f : \tilde{\Gamma} \rightarrow \Gamma$ may be extended to a cover $g : \tilde{\Delta} \rightarrow \Gamma$ in such a way that all deck transformations of $f$ are restrictions of deck transformations of $g$.

Equations

$$N(H)=\{a\in M:aH{a}^{-1},a^{-1}Ha\subseteq H\}$$

Full text

AMS subject numbers: 20M18, 57M10

Key words and phrases: graph immersion, deck transformation, free inverse monoid

1. Introduction

It is well known that group theory provides a powerful algebraic tool for studying covering spaces of topological spaces. For example, under mild conditions on a connected topological space $\mathcal{X}$, the connected covers of $\mathcal{X}$ may be classified via subgroups of the fundamental group of $\mathcal{X}$. This may be used to study deck transformations of covering spaces and actions of groups on topological spaces. However, the study of immersions between connected topological spaces seems to require somewhat different algebraic tools, even for graphs (1-dimensional $CW$-complexes).

In his paper [16], Stallings made use of immersions between finite graphs to study finitely generated subgroups of free groups. Here by an immersion between graphs we mean a locally injective graph morphism, that is, a graph morphism that is injective on star sets. Subsequently, Margolis and Meakin [7] showed how the theory of inverse monoids may be used to classify immersions between connected graphs. These results have been extended by Meakin and Szakács [9], [10] to classify immersions between higher dimensional cell complexes.

In the present paper we extend the ideas of [7] to show how inverse monoids may be used to study deck transformations of immersions between connected graphs. By a deck transformation we mean a graph automorphism that respects the immersion.

In Section 2 of the paper we introduce the terminology needed to describe covers and immersions of graphs. We then summarize some of the classical algebraic and topological ideas involving the classification of covers and the theory of deck transformations of connected covers of graphs.

Section 3 summarizes some of the basic theory of inverse monoids that will be needed subsequently. We then describe the use of closed inverse submonoids of free inverse monoids to classify immersions between connected graphs. We prove an apparently new result constructing the group of right $\omega$-cosets of a closed inverse submonoid of an inverse monoid in its normalizer.

In Section 4 we provide a calculation of the group of deck transformations of a connected immersion between graphs. If $H$ is the closed inverse submonoid of the free inverse monoid that is used to construct the immersion, then the group of deck transformations of the immersion is the group of right $\omega$-cosets of $H$ in its normalizer (Theorem 4.4).

In Section 5 we describe how the results of earlier sections of the paper specialize in the case that the graph immersion is actually a cover of connected graphs. In Section 6 we make an observation relating graph immersions to actions of groups on graphs. In Section 7 we prove that an immersion between graphs may be extended to a covering map between graphs in such a way that deck transformations of the immersion are restrictions of deck transformations of the cover.

2. Covers and immersions of graphs

By a graph ${\Gamma}=(\Gamma^{0},\Gamma^{1})$ we mean a graph in the sense of Serre [15]. Here $\Gamma^{0}$ is the set of vertices and $\Gamma^{1}$ is the set of edges of $\Gamma$. Thus every directed edge $e:v\rightarrow w$ comes equipped with an inverse edge $e^{-1}:w\rightarrow v$ such that $(e^{-1})^{-1}=e$ and $e^{-1}\neq e$. The initial vertex of $e$ is denoted by ${\alpha}(e)$ and the terminal vertex of $e$ is denoted by ${\omega}(e)$: thus $\alpha(e^{-1})=\omega(e)$ and $\omega(e^{-1})=\alpha(e)$. For each edge $e$ we designate one of the edges in the set $\{e,e^{-1}\}$ as being positively oriented, and its inverse edge as being negatively oriented. We normally only indicate the positively oriented edges in a sketch of a graph. A path in the graph $\Gamma$ is a finite string $p=e_{1}e_{2}...e_{n}$ where $\omega(e_{i})=\alpha(e_{i+1})$ for $i=1,...,n-1$: here the edges $e_{i}$ may be either positively or negatively oriented. We denote the initial vertex of $p$ by $\alpha(p)$ and the terminal vertex by $\omega(p)$: that is, if $p=e_{1}e_{2}...e_{n}$, then $\alpha(p)=\alpha(e_{1})$ and $\omega(p)=\omega(e_{n})$. The inverse of the path $p=e_{1}e_{2}...e_{n}$ is the path $p^{-1}=e_{n}^{-1}...e_{2}^{-1}e_{1}^{-1}$. The path $p$ is a circuit if $\alpha(p)=\omega(p)$. A tree is a connected graph in which every circuit $e_{1}e_{2}..e_{n}$ contains a subpath of the form $ee^{-1}$ for some edge $e$. Thus the Cayley graph $\Gamma(X)$ of the free group $FG(X)$ with respect to a set $X$ of free generators is a tree.

The free category on a graph $\Gamma$ is the category $FC(\Gamma)$ whose objects are the vertices of $\Gamma$ and whose morphisms are the paths in $\Gamma$. The product $p.q$ of paths $p$ and $q$ is defined in $FC(\Gamma)$ if and only if $\omega(p)=\alpha(q)$ and in that case $p.q=pq$, the concatenation of the path $p$ followed by the path $q$. We say that a path $p_{1}$ is an initial segment of a path $p$ if there is a path $p_{2}$ such that $p=p_{1}p_{2}$ and in this case $p_{2}$ is a terminal segment of $p$.

A morphism from the graph $\Gamma$ to the graph $\Gamma^{\prime}$ is a pair of functions $f:\Gamma\rightarrow\Gamma^{\prime}$ that takes vertices to vertices and edges to edges, and preserves incidence and orientation of edges. (Here we abuse notation slightly by using the same symbol $f$ to denote the corresponding function that takes vertices to vertices and the function that takes edges to edges.) If $v\in V({\Gamma})$, let $star({\Gamma},v)=\{e\in E({\Gamma}):{\alpha}(e)=v\}$. A morphism $f:{\Gamma}\rightarrow{\Gamma}^{\prime}$ induces a map $f_{v}:star({\Gamma},v)\rightarrow star({\Gamma}^{\prime},f(v))$ between star sets in the obvious way. Following Stallings [16], we say that a graph morphism $f$ is a cover if each $f_{v}$ is a bijection and that $f$ is an immersion if each $f_{v}$ is an injection.

In his paper [16], Stallings made use of immersions between graphs to study subgroups of free groups. Subsequently, Stallings foldings and Stallings graphs have been used extensively to study subgroups of free groups. See for example the survey paper by Kapovich and Myasnikov [4] or the paper by Birget, Margolis, Meakin and Weil [1] for just some of the relevant literature.

It is clear from the definition of a graph that it is possible to label the edges of a graph with labels of positively oriented edges coming from some set $X$ so that no two positively oriented edges with the same initial or terminal vertex are assigned the same label. The labeling of positively oriented edges may be extended to a labeling of all edges in the graph in such a way that if $e$ is a positively oriented edge labeled by $x\in X$ then $e^{-1}$ is labeled by $x^{-1}\in X^{-1}$, where $X^{-1}$ is a set disjoint from $X$ and in one-one correspondence with $X$ via the map $x\rightarrow x^{-1}$. We denote the label on an edge $e$ by $\ell(e)$. Thus $\ell(e)\in X$ if $e$ is a positively oriented edge and in general $\ell(e)\in X\cup X^{-1}$. The labeling of edges in $\Gamma$ extends to a labeling of paths in $\Gamma$ in the obvious way via $\ell(pq)=\ell(p)\ell(q)$ if $pq$ is defined. Thus if $p$ is a path in $\Gamma$ then $\ell(p)\in(X\cup X^{-1})^{*}$.

For example, the Cayley graph ${\Gamma}(G,X)$ of a group $G$ relative to a set $X$ of generators is obviously labeled over $X\cup X^{-1}$: its vertices are the elements of $G$ and there is a directed edge labeled by $x$ from $g$ to $gx$ for each $g\in G$ and each $x\in X\cup X^{-1}$. The bouquet of circles $B_{X}$ has one vertex and one positively oriented edge labeled by $x$ for each $x\in X$: of course $B_{X}$ also has negatively oriented edges labeled by elements $x^{-1}\in X^{-1}$ for each $x\in X$.

If $\Gamma$ is labeled over $X\cup X^{-1}$ as above, then the associated natural map $f_{\Gamma}$ from ${\Gamma}$ to $B_{X}$ that preserves edge labeling is a graph immersion. If $f:\Gamma\rightarrow\Gamma^{\prime}$ is a graph immersion and the edges of $\Gamma^{\prime}$ are labeled over $X\cup X^{-1}$ as above, then this labeling induces a labeling of the edges of $\Gamma$ in a natural way so that $f$ preserves edge labeling and $f_{\Gamma^{\prime}}\circ f=f_{\Gamma}$.

While the essential results in this paper may be formulated without resort to labeling the edges of our graphs, it is more convenient to do so. Hence we adopt the convention that all graphs that we will consider in this paper will be edge labeled as described above and all immersions will preserve edge labeling.

A simple example illustrating these ideas is provided in Figure 1: the natural map from the graph ${\Gamma}_{1}$ to $B_{\{a,b\}}$ is a cover, while the natural map from ${\Gamma}_{2}$ to $B_{\{a,b\}}$ is an immersion that is not a cover.

Figure 1

We now list some straightforward propositions that will be used in the sequel.

Proposition 2.1**.**

An immersion $f:\Gamma\rightarrow\Gamma^{\prime}$ between connected edge-labeled graphs is uniquely determined by the image $f(v)$ of any vertex $v$ in $\Gamma$. More precisely, if $v$ is a vertex of $\Gamma$ and $v^{\prime}$ is a vertex of $\Gamma^{\prime}$, then there is at most one graph immersion $f:\Gamma\rightarrow\Gamma^{\prime}$ such that $f(v)=v^{\prime}$. Such an immersion exists if and only if, for every path $p$ in $\Gamma$ with $\alpha(p)=v$, there is a path $p^{\prime}$ in $\Gamma^{\prime}$ with $\alpha(p^{\prime})=v^{\prime}$ and $\ell(p^{\prime})=\ell(p)$ and such that $p^{\prime}$ is a circuit if $p$ is a circuit.

Proof.

Suppose that there is an immersion $f$ from $\Gamma$ to $\Gamma^{\prime}$ such that $f(v)=v^{\prime}$. Let $v_{1}$ be any vertex of $\Gamma$ and $p$ any path in $\Gamma$ from $v$ to $v_{1}$. Since $f$ preserves edge labels, it maps paths in $\Gamma$ to paths in $\Gamma^{\prime}$ with the same label, so there must be a path $p^{\prime}$ in $\Gamma^{\prime}$ with $\alpha(p^{\prime})=v^{\prime}$ and $\ell(p^{\prime})=\ell(p)$. Furthermore, the path $p^{\prime}$ is unique since edge labeling is consistent with an immersion into $B_{X}$. It follows that we must have $f(v_{1})=\omega(p^{\prime})$. If $p$ is a circuit, then $\omega(p)=v$ so $\omega(p^{\prime})=f(\omega(p))=f(v)=v^{\prime}$, and so $p^{\prime}$ is a circuit. Also, if $e$ is an edge in $\Gamma$ with $\alpha(e)=v_{1}$, then we must have an edge $e^{\prime}$ in $\Gamma^{\prime}$ with $\alpha(e^{\prime})=\omega(p^{\prime})$ and $f(e)=e^{\prime}$. The uniqueness of such an edge follows since $\ell(e^{\prime})=\ell(e)$. So if there is an immersion from $\Gamma$ to $\Gamma^{\prime}$ that maps $v$ to $v^{\prime}$, there is only one such immersion.

Suppose conversely that for every path $p$ in $\Gamma$ with $\alpha(p)=v$, there is a path $p^{\prime}$ in $\Gamma^{\prime}$ with $\alpha(p^{\prime})=v^{\prime}$ and $\ell(p^{\prime})=\ell(p)$ and such that $p^{\prime}$ is a circuit if $p$ is a circuit. Let $v_{1}$ be a vertex in $\Gamma$ and $p$ a path from $v$ to $v_{1}$ in $\Gamma$. Then there is a (necessarily unique) path $p^{\prime}$ in $\Gamma^{\prime}$ with $\alpha(p^{\prime})=v^{\prime}$ and $\ell(p^{\prime})=\ell(p)$. Define $f(v_{1})=\omega(p^{\prime})$. If $p_{1}$ is another path in $\Gamma$ from $v$ to $v_{1}$, then as above, there is a unique path $p_{1}^{\prime}$ in $\Gamma^{\prime}$ with $\alpha(p_{1}^{\prime})=v^{\prime}$ and $\ell(p_{1}^{\prime})=\ell(p_{1})$. Since $p_{1}p^{-1}$ is a circuit in $\Gamma$ from $v$ to $v$ it follows by hypothesis that there is a circuit $q^{\prime}$ in $\Gamma^{\prime}$ at $v^{\prime}$ with $\ell(q^{\prime})=\ell(p_{1}p^{-1})=\ell(p_{1})\ell(p^{-1})=\ell(p_{1}^{\prime})\ell(p^{\prime-1})$. So $p_{1}^{\prime}$ is an initial segment of $q^{\prime}$ and $p^{\prime-1}$ is a terminal segment of $q^{\prime}$, and hence $p_{1}^{\prime}p^{\prime-1}$ is a circuit in $\Gamma^{\prime}$ from $v^{\prime}$ to $v^{\prime}$ with $\ell(p_{1}^{\prime}p^{\prime-1})=\ell(q^{\prime})$. It follows that $\omega(p_{1}^{\prime})=\omega(p^{\prime})$, so $f$ is well defined on vertices. If $e$ is an edge in $\Gamma$ with $\alpha(e)=v_{1}$, then $pe$ is a path in $\Gamma$ starting at $v$ so there is some path $s^{\prime}$ in $\Gamma^{\prime}$ starting at $v^{\prime}$ with $\ell(s^{\prime})=\ell(pe)=\ell(p)\ell(e)=\ell(p^{\prime})\ell(e)$. By uniqueness of a path starting at $v^{\prime}$ with this label and the fact that $p^{\prime}$ is a path starting at $v^{\prime}$ with $\ell(p^{\prime})=\ell(p)$, it follows that there must be a (unique) edge $e^{\prime}$ in $\Gamma^{\prime}$ with $\alpha(e^{\prime})=\omega(p^{\prime})$ and $\ell(e^{\prime})=\ell(e)$. Then define $f(e)=e^{\prime}$. This is well defined by uniqueness of the edge $e^{\prime}$. So $f$ is a well defined morphism from $\Gamma$ to $\Gamma^{\prime}$ which is clearly an immersion since it preserves edge labeling. ∎

An isomorphism from the edge labeled graph $\Gamma$ onto the edge labeled graph $\Gamma^{\prime}$ is a (label-preserving) graph morphism that is a bijection from vertices of $\Gamma$ to vertices of $\Gamma^{\prime}$ and also a bijection from edges of $\Gamma$ to edges of $\Gamma^{\prime}$. Such an isomorphism must restrict to isomorphisms between the connected components of $\Gamma$ and the connected components of $\Gamma^{\prime}$ and, by Proposition 2.1, it is determined by the image of any vertex in $\Gamma$ if $\Gamma$ is connected. The following fact is an easy consequence of Proposition 2.1.

Proposition 2.2**.**

Let $\Gamma$ and $\Gamma^{\prime}$ be (edge labeled) connected graphs, $v$ a vertex in $\Gamma$ and $v^{\prime}$ a vertex in $\Gamma^{\prime}$. Then a bijective map $f:\Gamma\rightarrow\Gamma^{\prime}$ such that $f(v)=v^{\prime}$ is an isomorphism from $\Gamma$ onto $\Gamma^{\prime}$ if and only if there is a label-preserving bijection $p\leftrightarrow p^{\prime}$ between the paths $p$ in $\Gamma$ starting at $v$ and the paths $p^{\prime}$ in $\Gamma^{\prime}$ starting at $v^{\prime}$ such that $p$ is a circuit at $v$ if and only if $p^{\prime}$ is a circuit at $v^{\prime}$.

Let $f:\tilde{\Gamma}\rightarrow\Gamma$ be an immersion of connected graphs, $v$ a vertex in $\Gamma$ and $\tilde{v}\in f^{-1}(v)$. Then we say that a path $p$ in $\Gamma$ starting at $v$ lifts to $\tilde{v}$ if there is a path $\tilde{p}$ in $\tilde{\Gamma}$ starting at $\tilde{v}$ such that $f(\tilde{p})=p$. In this case it is clear that the lifted path $\tilde{p}$ is uniquely determined by $\tilde{v}$ and $p$ since $\ell(\tilde{p})=\ell(p)$. We refer to the proposition below as the “path lifting” proposition.

Proposition 2.3**.**

Let $f:\tilde{\Gamma}\rightarrow\Gamma$ be an immersion between connected graphs. Then $f$ is a cover if and only if, for each vertex $v$ in $\Gamma$ and each vertex $\tilde{v}$ in $f^{-1}(v)$, every path $p$ in $\Gamma$ starting at $v$ lifts to a (unique) path in $\tilde{\Gamma}$ starting at $\tilde{v}$.

Proof.

If $f$ is a cover, then the fact that every path $p$ lifts to all preimages of $v$ is easy to prove and may be viewed as a very special case of a much more general path lifting property for covers of topological spaces (see for example Proposition 1.30 of Hatcher’s book [3]). Conversely, suppose that every path in $\Gamma$ starting at $v$ lifts to a path in $\tilde{\Gamma}$ starting at $\tilde{v}$. It follows that if $e$ is an edge in $\Gamma$ with $\alpha(e)=v$, then there is an edge $\tilde{e}$ in $\tilde{\Gamma}$ with $\alpha(\tilde{e})=\tilde{v}$ and $f(\tilde{e})=e$. So the map $f_{\tilde{v}}$ from $star(\tilde{\Gamma},\tilde{v})$ to $star(\Gamma,v)$ is surjective. Since it is also injective by hypothesis, it is a bijection, and so the immersion $f$ is a cover. ∎

The path lifting proposition above does not hold for immersions that are not covers in general, but it is easy to see that maximal initial segments of paths in $\Gamma$ lift uniquely to paths in $\tilde{\Gamma}$, as described in the following proposition.

Proposition 2.4**.**

Let $f:\tilde{\Gamma}\rightarrow\Gamma$ be an immersion between connected graphs, let $v$ be a vertex of $\Gamma$ and let $p$ be a path in $\Gamma$ with ${\alpha}(p)=v$. Then for every vertex $\tilde{v}\in f^{-1}(v)$ there is a unique (possibly empty) maximal initial segment $p_{1}$ of $p$ that lifts to a path at $\tilde{v}$. Furthermore, the lift of $p_{1}$ at $\tilde{v}$ is unique.

Proof.

Since an immersion is locally injective on star sets, it is clear that an edge $e$ of $\Gamma$ starting at $v$ lifts to at most one edge $\tilde{e}$ in $\tilde{\Gamma}$ starting at $\tilde{v}$. The result then follows by an easy inductive argument. ∎

We briefly summarize some of the most basic facts linking group theory and covers of graphs. Recall (see Stallings [16]) that two paths $p$ and $q$ in a graph $\Gamma$ are said to be homotopy equivalent (written $p\sim q$) if and only if it is possible to pass from $p$ to $q$ by a finite sequence of insertions or deletions of paths of the form $ee^{-1}$ for various edges $e$ of $\Gamma$. Clearly $\alpha(p)=\alpha(q)$ and $\omega(p)=\omega(q)$ if $p\sim q$. We denote the equivalence class (homotopy class) of a path $p$ in $\Gamma$ by $[p]$. The *fundamental groupoid * $\pi_{1}(\Gamma)$ is a groupoid whose objects are the vertices of $\Gamma$ and whose morphisms are the homotopy classes $[p]$. We regard $[p]$ as a morphism from $\alpha(p)$ to $\omega(p)$. The multiplication in the groupoid is defined by $[p][q]=[pq]$ if $\omega(p)=\alpha(q)$ and is undefined otherwise.

For each vertex $v$ of $\Gamma$ the set $\pi_{1}(\Gamma,v)=\{[p]:p$ is a circuit from $v$ to $v$ in $\Gamma\}$ is a group with respect to the multiplication in $\pi_{1}(\Gamma)$, called the fundamental group of $\Gamma$ based at $v$. The following fact is classical and can be found in many sources, for example [16].

Proposition 2.5**.**

Let $\Gamma$ be a connected graph and $v,w$ vertices of $\Gamma$. Then

(a) $\pi_{1}(\Gamma,v)$ is a free group whose rank is the number of positively oriented edges of $\Gamma$ that are not in a spanning tree for $\Gamma$.

(b) If $p$ is a path in $\Gamma$ from $v$ to $w$, then the map $[q]\rightarrow[p][q][p^{-1}]$ for $[q]\in\pi_{1}(\Gamma,w)$ defines an isomorphism from $\pi_{1}(\Gamma,w)$ onto $\pi_{1}(\Gamma,v)$.

It is well known (see for example Hatcher’s book [3]) that under suitable conditions on a connected topological space $\mathcal{X}$, connected covers of $\mathcal{X}$ may be classified via conjugacy classes of subgroups of the fundamental group of $\mathcal{X}$. The following version of this result for graph covers may be found in many sources, for example, [3] or [16].

Theorem 2.6**.**

(a) Let $f:\tilde{\Gamma}\rightarrow\Gamma$ be a cover of connected graphs, let $v$ be a vertex of $\Gamma$ and $\tilde{v}\in f^{-1}(v)$. Then $f$ induces an embedding of $\pi_{1}(\tilde{\Gamma},\tilde{v})$ into $\pi_{1}(\Gamma,v)$. If $\tilde{v_{1}}$ is another vertex in $f^{-1}(v)$, then the groups $f(\pi_{1}(\tilde{\Gamma},\tilde{v}))$ and $f(\pi_{1}(\tilde{\Gamma},\tilde{v_{1}}))$ are conjugate subgroups of $\pi_{1}(\Gamma,v)$.

(b) Conversely, let $\Gamma$ be a connected graph, $v$ a vertex in $\Gamma$ and $H\leq\pi_{1}(\Gamma,v)$. Then there is a unique (up to labeled graph isomorphism) connected graph $\tilde{\Gamma}$ and a unique (up to equivalence) covering map $f:\tilde{\Gamma}\rightarrow\Gamma$ and a vertex $\tilde{v}\in\tilde{\Gamma}$ such that $f(\tilde{v})=v$ and $f(\pi_{1}(\tilde{\Gamma},\tilde{v}))=H$.

If $f:\tilde{\Gamma}\rightarrow\Gamma$ is an immersion of connected graphs, a labeled graph automorphism $\gamma$ of $\tilde{\Gamma}$ is called a deck transformation of $\tilde{\Gamma}$ if $f=f\circ\gamma$, i.e. $f(\tilde{v})=f(\gamma(\tilde{v}))$ for all vertices $\tilde{v}$ in $\tilde{\Gamma}$. The deck transformations of $\tilde{\Gamma}$ form a group $G(\tilde{\Gamma})$ with respect to composition of automorphisms.

A graph cover $f:\tilde{\Gamma}\rightarrow\Gamma$ is called a normal cover if, for every vertex $v$ in $\Gamma$ and every pair of vertices $\tilde{v}_{1},\tilde{v}_{2}\in f^{-1}(v)$, there is a deck transformation that takes $\tilde{v}_{1}$ to $\tilde{v}_{2}$: equivalently ([3], Proposition 1.39), $f$ is a normal cover if and only if the subgroup $H=f(\pi_{1}(\tilde{\Gamma},\tilde{v}))$ of $\pi_{1}(\Gamma,f(v))$ that defines the cover is a normal subgroup of $\pi_{1}(\Gamma,f(v))$. The universal cover of $\Gamma$ is the cover $\tilde{\Gamma}$ corresponding to the trivial subgroup of $\pi_{1}(\Gamma,f(v))$: a cover of $\Gamma$ is isomorphic to the universal cover if and only if it is a tree.

If $f:\tilde{\Gamma}\rightarrow\Gamma$ is a cover of connected graphs, then there is a well known connection between the group $G(\tilde{\Gamma})$ of deck transformations and the fundamental group of $\Gamma$. The following result is a special case of a more general standard result in topology (see for example [3], Proposition 1.39).

Theorem 2.7**.**

Let $f:\tilde{\Gamma}\rightarrow\Gamma$ be a cover of connected graphs, let $\tilde{v}$ be a vertex of $\tilde{\Gamma}$ with $f(\tilde{v})=v$ and let $H$ be the subgroup $H=f(\pi_{1}(\tilde{\Gamma},\tilde{v}))$ of $\pi_{1}(\Gamma,v)$. Then $G(\tilde{\Gamma})\cong N(H)/H$, where $N(H)$ is the normalizer of $H$ in $\pi_{1}(\Gamma,v)$. In particular, if $\tilde{\Gamma}$ is the universal cover of $\Gamma$, then $G(\tilde{\Gamma})\cong\pi_{1}(\Gamma,v)$.

While group theory provides a powerful algebraic tool for classifying and studying covers of graphs (or topological spaces in general), it appears that groups do not provide an adequate algebraic tool to classify immersions between graphs. For example let $\Gamma(a)$ denote the Cayley graph of $\mathbb{Z}=Gp\langle a:\emptyset\rangle$ with respect to the generating set $\{a\}$. The vertices of $\Gamma(a)$ may be identified with the integers and there is a directed edge labeled by $a$ from $n$ to $n+1$ for each integer $n$. Then $\Gamma(a)$ is the universal cover of the circle $B_{\{a\}}$. Any connected subgraph of $\Gamma(a)$ immerses into $B_{\{a\}}$ but all such graphs have trivial fundamental groups, so they cannot be distinguished by subgroups of $\mathbb{Z}$. We need a different algebraic tool to classify immersions and to encode the fact that paths only sometimes lift under immersions (Proposition 2.4). In subsequent sections of this paper, we show that the theory of inverse monoids provides a useful algebraic tool to study graph immersions and in particular to obtain analogues of Theorems 2.6 and 2.7.

3. Inverse monoids and their closed inverse submonoids

An inverse monoid is a monoid $M$ such that for every element $a\in M$ there is a unique element $a^{-1}$ in $M$ such that

$a=aa^{-1}a\,\,\,\,$ and $\,\,\,\,a^{-1}=a^{-1}aa^{-1}.$

It is clear that the elements $aa^{-1}$ and $a^{-1}a$ of an inverse monoid are idempotents of $M$. In general, $aa^{-1}\neq a^{-1}a$ and neither of these idempotents is necessarily equal to the identity $1$ of $M$. We denote the set of idempotents of an inverse monoid $M$ by $E(M)$. An important elementary fact about inverse monoids is that their idempotents commute, i.e. $ef=fe$ for all $e,f\in M$. In fact inverse monoids may be characterized alternatively as (von Neumann) regular monoids whose idempotents commute.

Inverse monoids provide an appropriate algebraic tool for studying partial symmetry of mathematical objects in much the same way as groups are used to study symmetry. We refer the reader to the book of Lawson [5] for an exposition of this point of view and for much basic information about inverse monoids.

A standard example of an inverse monoid is the symmetric inverse monoid $SIM(X)$ on a set $X$. This is the set of bijections between subsets of $X$ with respect to the usual composition of partial maps. The inverse of a bijection $f\in SIM(X)$ with domain $A$ and range $B$ is the inverse map $f^{-1}$ with domain $B$ and range $A$: the idempotents of $SIM(X)$ are the identity maps on subsets of $X$ (including the empty map [math] with domain the empty subset). The analogue for inverse monoids of Cayley’s theorem for groups is the Wagner-Preston Theorem, namely every inverse monoid embeds in a suitable symmetric inverse monoid (see [5] for a proof of this theorem).

The natural partial order on an inverse monoid $M$ is defined by $a\leq b$ iff $a=eb$ for some idempotent $e\in E(M)$ (equivalently $a=bf$ for some idempotent $f\in E(M)$ or equivalently $a=aa^{-1}b$ or equivalently $a=ba^{-1}a$). This extends the natural partial order on $E(M)$ defined by $e\leq f$ iff $e=ef=fe$. With respect to this partial order, $E(M)$ forms a lower semilattice with meet operation $e\wedge f=ef$ for all $e,f\in E(M)$. For the symmetric inverse monoid $SIM(X)$, the natural partial order corresponds to restriction of a partial one-one map to a subset of its domain. The semilattice of idempotents of $SIM(X)$ is of course the Boolean lattice of subsets of $X$ with respect to inclusion.

For each subset $N$ of an inverse monoid $M$, we denote by $N^{\omega}$ the set of all elements $m\in M$ such that $m\geq n$ for some $n\in N$. The subset $N$ of $M$ is called closed if $N=N^{\omega}$. Closed inverse submonoids of an inverse monoid $M$ arise naturally in the representation theory of $M$ by partial injections on a set developed by Schein [14]. An inverse monoid $M$ acts by injective partial functions on a set $Q$ if there is a homomorphism from $M$ to $SIM(Q)$. Denote by $qm$ the image of $q$ under the action of $m$ if $q$ is in the domain of the action by $m$. (Here we are considering $M$ as acting on the right on $Q$.)

If an inverse monoid $M$ acts on $Q$ by injective partial functions, then for every $q\in Q,\,Stab(q)=\{m\in M:qm=q\}$ is a closed inverse submonoid of $M$. Conversely, given a closed inverse submonoid $H$ of $M$, we can construct a transitive representation of $M$ as follows. A subset of $M$ of the form $(Hm)^{\omega}$ where $mm^{-1}\in H$ is called a right ${\omega}$-coset of $H$. Let $X_{H}$ denote the set of right $\omega$-cosets of $H$. If $(Hn)^{\omega}\in X_{H}$ and $m\in M$, define an action by $(Hn)^{\omega}m=(Hnm)^{\omega}$ if $(Hnm)^{\omega}\in X_{H}$ and undefined otherwise. This defines a transitive action of $M$ (on the right) on $X_{H}$. Conversely, if $M$ acts transitively on $Q$, then this action is equivalent in the obvious sense to the action of $M$ on the right $\omega$-cosets of $Stab(q)$ in $M$ for any $q\in Q$. See [14] or [12] for details. Dually, left $\omega$-cosets of $H$ in $M$ (sets of the form $(mH)^{\omega}$ where $m^{-1}m\in H$) arise in connection with left actions of $M$ by partial one-one maps on some set.

If $M$ is generated as an inverse monoid by a set $X$ and $H$ is a closed inverse submonoid of $M$ then the graph of right $\omega$-cosets of $H$ in $M$ is the graph with vertices the set $X_{H}$ of right $\omega$-cosets of $H$ in $M$ and with an edge labeled by $x\in X\cup X^{-1}$ from $(Hm)^{\omega}$ to $(Hmx)^{\omega}$ if $mm^{-1},mxx^{-1}m^{-1}\in H$.

It is well known that free inverse monoids exist. We will denote the free inverse monoid on a set $X$ by $FIM(X)$. The structure of free inverse monoids on one generator was determined by Gluskin [2]. The structure of free inverse monoids in general was determined much later independently by Scheiblich [13] and Munn [11]. Scheiblich’s description for elements of $FIM(X)$ is in terms of rooted Schreier subsets of the free group $FG(X)$, while Munn’s description is in terms of birooted edge-labeled trees. Scheiblich’s description provides an important example of a McAlister triple, in the spirit of the McAlister $P$-theorem [8], while Munn’s description lends itself most directly to a solution to the word problem for $FIM(X)$. It is not difficult to see the equivalence of the two descriptions. A variation on Scheiblich’s approach is provided in Lawson’s book [5]. The version below is a slight variation on Munn’s approach, the essential difference being that for some purposes it is somewhat more convenient to regard Munn’s birooted trees as subtrees of the Cayley graph of the free group $FG(X)$, with the initial root identified with the vertex $1$ in the Cayley graph.

Denote by ${\Gamma}(X)$ the Cayley graph of the free group $FG(X)$ with respect to the usual presentation, $FG(X)=Gp\langle X:\emptyset\rangle$. Thus ${\Gamma}(X)$ is an infinite tree whose vertices correspond to the elements of $FG(X)$ (in reduced form) and with a directed edge labeled by $x\in X$ from $g$ to $gx$ (and an inverse edge labeled by $x^{-1}$ from $gx$ to $g$). For each word $w\in(X\cup X^{-1})^{*}$, denote by $MT(w)$ the finite subtree of the tree ${\Gamma}(X)$ obtained by reading the word $w$ as the label of a path in ${\Gamma}(X)$, starting at $1$. Thus, for example, if $w=aa^{-1}bb^{-1}ba^{-1}abb^{-1}$, then $MT(w)$ is the tree pictured in Figure 2.

Figure 2

One may view $MT(w)$ as a birooted tree, with initial root $1$ and terminal root $r(w)$, the reduced form of the word $w$ in the usual group-theoretic sense. Munn’s solution [11] to the word problem in $FIM(X)$ may be stated in the following form.

Theorem 3.1**.**

If $u,v\in(X\cup X^{-1})^{*}$, then $u=v$ in $FIM(X)$ iff $MT(u)=MT(v)$ and $r(u)=r(v)$.

Thus elements of $FIM(X)$ may be viewed as pairs $(MT(w),r(w))$ (or as birooted edge-labeled trees, which was the way that Munn described his results). Multiplication in $FIM(X)$ is performed as follows. If $u,v\in(X\cup X^{-1})^{*}$, then $MT(uv)=MT(u)\cup r(u).MT(v)$ (just translate $MT(v)$ so that its initial root coincides with the terminal root of $MT(u)$ and take the union of $MT(u)$ and the translated copy of $MT(v)$: the terminal root is of course $r(uv)$). The idempotents of $FIM(X)$ are the Dyck words, i.e. the words $w$ whose reduced form is $1$: two Dyck words represent the same idempotent of $FIM(X)$ if and only if they have the same Munn tree. Munn’s approach has been greatly extended by Stephen [17] to a general theory of presentations of inverse monoids by generators and relations.

In their paper [7], Margolis and Meakin studied closed inverse submonoids of free inverse monoids and showed how they could be used to classify immersions between connected graphs. In particular, they showed that closed inverse submonoids of free inverse monoids have surprisingly nice finiteness properties, and they may be constructed from free actions of groups on trees. A closed inverse submonoid of a free inverse monoid is not necessarily a free inverse monoid, but it admits an idempotent pure morphism onto a free inverse monoid. We briefly recall some of the results from the paper [7] that are relevant for our current purposes.

Recall that an inverse category $\mathcal{C}$ is a category with the property that for each morphism $p$ there is a unique morphism $p^{-1}$ such that $p=pp^{-1}p$ and $p^{-1}=p^{-1}pp^{-1}$. The loop monoid of such a category at the vertex (object) $v$ is the set $L({\mathcal{C}},v)=\{p:p$ is a morphism in $\mathcal{C}$ from $v$ to $v\}$. Then $L({\mathcal{C}},v)$ is an inverse monoid. Let $\Gamma$ be a (connected) graph. Recall from [7] that the free inverse category $FIC(\Gamma)$ on $\Gamma$ is the quotient of the free category $FC(\Gamma)$ on $\Gamma$ by the category congruence $\sim_{i}$ induced by all relations of the form $p=pp^{-1}p$ and $pp^{-1}qq^{-1}=qq^{-1}pp^{-1}$ if $\alpha(p)=\alpha(q)$ for paths $p,q$ in $\Gamma$. Denote the $\sim_{i}$-class of $p$ by $[[p]]$. Then $FIC(\Gamma)$ is an inverse category. Of course if $\Gamma=B_{X}$ then $FIC(\Gamma)=FIM(X)$.

The loop monoid of $FIC(\Gamma)$ at the vertex $v$ of $\Gamma$ is

$L(\Gamma,v)=\{[[p]]:p$ is a circuit in $\Gamma$ based at $v\}$.

If $\Gamma$ is labeled over $X\cup X^{-1}$ consistent with an immersion into $B_{X}$ it follows that if $p\,\sim_{i}\,q$ for paths $p,q$ in $\Gamma$, then $\ell(p)$ and $\ell(q)$ are equal in $FIM(X)$. So we will view labels of paths in $\Gamma$ as elements of $FIM(X)$ throughout the sequel. We note that $FIM(X)$ acts on $\Gamma^{0}$ in a natural way (namely $w\in FIM(X)$ acts on $v_{1}$ and takes $v_{1}$ to $v_{2}$ if there is a path labeled by $w$ from $v_{1}$ to $v_{2}$). Then $L(\Gamma,v)$ is the stabilizer of $v$ under this action, so each loop monoid is a closed inverse submonoid of $FIM(X)$. See Proposition 4.3 of [7] for details.

Recall that two closed inverse submonoids $H$ and $K$ of an inverse monoid $M$ are said to be conjugate (written $H\approx K$) if there exists $m\in M$ such that $mHm^{-1}\subseteq K$ and $m^{-1}Km\subseteq H$. Conjugation is an equivalence relation on the set of closed inverse submonoids of $M$. We caution however that, unlike the situation in group theory, conjugate closed inverse submonoid of inverse monoids are not necessarily isomorphic. For example, the subsets $\{1,aa^{-1},a^{2}a^{-2}\}$ and $\{1,aa^{-1},a^{-1}a,aa^{-2}a\}$ of $FIM(\{a\})$ are conjugate closed inverse submonoids of $FIM(\{a\})$ that are clearly not isomorphic.

Immersions of connected graphs over $B_{X}$ are classified via conjugacy classes of closed inverse submonoids of $FIM(X)$ as indicated in the following theorem (Theorem 4.4 of [7]).

Theorem 3.2**.**

Let $\Gamma$ be a connected graph with edges labeled over $X\cup X^{-1}$ consistent with an immersion into $B_{X}$. Then each loop monoid is a closed inverse submonoid of $FIM(X)$ and the set of all loop monoids $L(\Gamma,v)$ for $v$ a vertex of $\Gamma$ is a conjugacy class of the set of closed inverse submonoids of $FIM(X)$. Conversely, if $H$ is any closed inverse submonoid of a free inverse monoid $FIM(X)$ then there is some graph $\Gamma$ and an immersion $f:\Gamma\rightarrow B_{X}$ such that $H$ is a loop monoid of $FIC(\Gamma)$: furthermore, $\Gamma$ is unique (up to graph isomorphism) and $f$ is unique (up to equivalence).

The results of this theorem can be extended somewhat to obtain a classification of immersions over arbitrary graphs, as indicated in the following theorem (Theorem 4.5 of [7]). This theorem may be viewed as the analogue for graph immersions of the classification theorem for graph covers (Theorem 2.6 above).

Theorem 3.3**.**

Let $f:\Delta\rightarrow\Gamma$ be an immersion of connected graphs where $\Delta$ and $\Gamma$ are edge labeled over $X\cup X^{-1}$ consistent with immersions into $B_{X}$. If $v$ is a vertex of $\Gamma$ and $v_{1}\in f^{-1}(v)$ then $f$ induces an embedding of $L(\Delta,v_{1})$ into $L(\Gamma,v)$. Conversely, let $\Gamma$ be a graph edge-labeled over $X\cup X^{-1}$ as usual and let $H$ be a closed inverse submonoid of $FIM(X)$ so that $H\subseteq L(\Gamma,v)$ for some vertex $v$ in $\Gamma$. Then there is a graph $\Delta$, an immersion $f:\Delta\rightarrow\Gamma$ and a vertex $v_{1}$ in $\Delta$ such that $f(v_{1})=v$ and $f(L(\Delta,v_{1}))=H$. Furthermore, $\Delta$ is unique (up to graph isomorphism) and $f$ is unique (up to equivalence). If $H,K$ are two closed inverse submonoids of $FIM(X)$ with $H,K\subseteq L(\Gamma,v)$, then the corresponding immersions $f:\Delta\rightarrow\Gamma$ and $g:\Delta^{\prime}\rightarrow\Gamma$ are equivalent iff $H\approx K$ in $FIM(X)$.

Remark From the proof of Theorem 3.3 in [7], it follows that the graph $\Delta$ constructed in this theorem is the graph of right $\omega$-cosets of $H$ in $FIM(X)$.

Theorems 3.2 and 3.3 have been extended to classify immersions between $2$-dimensional $CW$-complexes and more generally between finite dimensional $\Delta$-complexes in the papers by Meakin and Szakács [9] and [10]. In these more general cases it is necessary to construct closed inverse submonoids of certain inverse monoids presented by generators and relations associated with the complexes.

We close this section with a result about normalizers of closed inverse submonoids of inverse monoids. Part (a) of Proposition 3.4 below was observed in a paper by Lawson, Margolis and Steinberg ([6], Lemma 2.10), but as far as we know the other parts of the proposition are new. This result will be needed in the construction of the group of deck transformations of a graph immersion later in this paper.

Let $H$ be a closed inverse submonoid of an inverse monoid $M$. Define the normalizer $N(H)$ of $H$ in $M$ to be the set

[TABLE]

Proposition 3.4**.**

Let $H$ be a closed inverse submonoid of an inverse monoid $M$. Then

1. (a)

$N(H)$* is a closed inverse submonoid of $M$ and $H$ is a full inverse submonoid of $N(H)$ (i.e. $H$ contains all of the idempotents of $N(H)$);* 2. (b)

If $a\in N(H)$ then the right ${\omega}$-coset $(Ha)^{\omega}$ and the left ${\omega}$-coset $(aH)^{\omega}$ both exist and $(Ha)^{\omega}=(aH)^{\omega}$; 3. (c)

The relation $\rho_{H}$ on $N(H)$ defined by $a\,\rho_{H}\,c$ iff $(Ha)^{\omega}=(Hc)^{\omega}$ is a congruence on $N(H)$ and $N(H)/{\rho_{H}}$ is a group with operation $(Ha)^{\omega}.(Hb)^{\omega}=(Hab)^{\omega}$.

Proof.

(a) It is clear that $a^{-1}\in N(H)$ if $a\in N(H)$ and also that $N(H)$ contains the identity of $M$. Also if $a,b\in N(H)$ then $(ab)^{-1}H(ab)=b^{-1}a^{-1}Hab\subseteq b^{-1}Hb\subseteq H$ and similarly $(ab)H(ab)^{-1}\subseteq H$. Thus $ab\in N(H)$, so $N(H)$ is an inverse submonoid of $FIM(X)$. If $a\in N(H)$ and $b\geq a$ then $a=be$ for some idempotent $e\in M$. Hence if $h\in H$ then $beheb^{-1}=aha^{-1}\in H$. Hence $bhb^{-1}\in H$ since $H$ is closed and $bhb^{-1}\geq beheb^{-1}$. Thus $bHb^{-1}\subseteq H$: similarly $b^{-1}Hb\subseteq H$. Hence $b\in N(H)$ and so $N(H)$ is a closed inverse submonoid of $M$. It is clear that $H\subseteq N(H)$. If $e$ is an idempotent of $N(H)$, then $e=e1e^{-1}\in H$ since $1\in H$ so $H$ is a full inverse submonoid of $N(H)$.

(b) If $a\in N(H)$ then $aa^{-1},\,a^{-1}a\in H$ since $H$ has an identity, so both left and right $\omega$-cosets exist. Note that if $a\in N(H)$ then $aHa^{-1}\subseteq H$ and so $a^{-1}aHa^{-1}a\subseteq a^{-1}Ha\subseteq H$ so via conjugation by $a$ we get $aHa^{-1}=aa^{-1}aHa^{-1}aa^{-1}\subseteq aa^{-1}Haa^{-1}\subseteq aHa^{-1}$. It follows that $aHa^{-1}=aa^{-1}Haa^{-1}$ and so $aHa^{-1}a=aa^{-1}Ha$.

Let $x\in(Ha)^{\omega}$, so $x\geq ha$, for some $h\in H$. Hence $x\geq aa^{-1}ha=ah_{1}a^{-1}a$ for some $h_{1}\in H$ by the observation above. Since $h_{2}=h_{1}a^{-1}a\in H$, and $x\geq ah_{2}$ it follows that $x\in(aH)^{\omega}$. Hence $(Ha)^{\omega}\subseteq(aH)^{\omega}$. A similar argument shows the converse inclusion.

(c) If $a\in N(H)$ then by part (b), $(Ha)^{\omega}$ exists and clearly $a\in(Ha)^{\omega}$. This shows that $\rho_{H}$ is a reflexive relation on $N(H)$. Since $\rho_{H}$ is obviously symmetric and transitive, it is an equivalence relation on $N(H)$. It is well known that $(Ha)^{\omega}$=$(Hc)^{\omega}$ if and only if $ac^{-1}\in H$ (see, for example, [12], Lemma IV.4.5.). Suppose $a\,\rho_{H}\,c$ and $b\,\rho_{H}\,d$, so that (Ha)^{\omega}$$=$$(Hc)^{\omega} and $(Hb)^{\omega}$=$(Hd)^{\omega}$. Then $ac^{-1},bd^{-1}\in H$. We have $abd^{-1}c^{-1}\in aHc^{-1}$, so $abd^{-1}c^{-1}=ahc^{-1}$, for some $h\in H$. But $ah\in(aH)^{\omega}=(Ha)^{\omega}$ since $a\in N(H)$, so $ah\geq h_{1}a$ for some $h_{1}\in H$. Hence $abd^{-1}c^{-1}\geq h_{1}ac^{-1}\in H$ and so $abd^{-1}c^{-1}\in H$. Hence $(Hab)^{\omega}$ = $(Hcd)^{\omega}$, i.e. $\rho_{H}$ is a congruence on $N(H)$.

The operation of congruence classes in $N(H)/{\rho_{H}}$ is of course $(Ha)^{\omega}.(Hb)^{\omega}=(Hab)^{\omega}$. Clearly $H$ = $(H1)^{\omega}$ is a right $\omega$-coset. It follows that, by definition, $(H1)^{\omega}$ . $(Ha)^{\omega}$ = $(Ha)^{\omega}$ = $(Ha)^{\omega}$ . $(H1)^{\omega}$, so $H$ acts as an identity. Also, if $a\in N(H)$, then $a^{-1}\in N(H)$ and $(Ha)^{\omega}$ . $(Ha^{-1})^{\omega}$ = $(Haa^{-1})^{\omega}$ = $H$ since $aa^{-1}\in H$. Similarly $(Ha^{-1})^{\omega}$ . $(Ha)^{\omega}$ = $H$, so $N(H)/{\rho_{H}}$ is a group. ∎

We denote the group $N(H)/{\rho_{H}}$ by $N(H)/H$ and refer to it as the group of $\omega$-cosets of $H$ in $N(H)$.

4. The group of deck transformations

Throughout this section $f:\tilde{\Gamma}\rightarrow\Gamma$ is an immersion of connected graphs, $v_{0}$ is a fixed basepoint in $\Gamma$, $\tilde{v}_{0}\in f^{-1}(v_{0})$ is a fixed basepoint in $\tilde{\Gamma}$ and $H=f(L(\tilde{\Gamma},\tilde{v}_{0}))$, a closed inverse submonoid of $L(\Gamma,v_{0})$.

A partial isomorphism of $\tilde{\Gamma}$ is a (labeled graph) isomorphism $\phi:\Delta_{1}\rightarrow\Delta_{2}$ between subgraphs $\Delta_{1}$ and $\Delta_{2}$ of $\tilde{\Gamma}$ that respects the immersion, that is $f(\tilde{v})=f(\phi(\tilde{v}))$ for all vertices $\tilde{v}$ in $\Delta_{1}$: it follows that $f(\tilde{e})=f(\phi(\tilde{e}))$ for all edges $\tilde{e}$ in $\Delta_{1}$. It is convenient to also consider the empty map from $\emptyset$ to $\emptyset$ as a partial isomorphism: this is denoted by [math]. We denote the set of partial isomorphisms of $\tilde{\Gamma}$ by $PI(\tilde{\Gamma})$. Partial isomorphisms may be composed in the usual way. Namely, if $\phi_{1}:\Delta_{1}\rightarrow\Delta_{2}$ and $\phi_{2}:\Delta_{3}\rightarrow\Delta_{4}$ are partial isomorphisms, then $\phi_{2}\circ\phi_{1}$ is the corresponding partial isomorphism from $\phi_{1}^{-1}(\Delta_{2}\cap\Delta_{3})$ to $\phi_{2}(\Delta_{2}\cap\Delta_{3})$ defined by $(\phi_{2}\circ\phi_{1})(\tilde{e})=\phi_{2}(\phi_{1}(\tilde{e}))$ for all vertices and edges $\tilde{e}\in\phi_{1}^{-1}(\Delta_{2}\cap\Delta_{3})$.

Proposition 4.1**.**

With respect to the multiplication above, $PI(\tilde{\Gamma})$ is an inverse monoid. The idempotents of $PI(\tilde{\Gamma})$ are the identity automorphisms on subgraphs of $\tilde{\Gamma}$ and the corresponding maximal subgroups are isomorphic to the group of deck transformations of the subgraph. In particular the group of units of $PI(\tilde{\Gamma})$ is the group $G(\tilde{\Gamma})$ of deck transformations of $\tilde{\Gamma}$. The natural partial order on $\tilde{\Gamma}$ is defined by $\phi_{1}\leq\phi_{2}$ if the domain of $\phi_{1}$ is a subgraph of the domain of $\phi_{2}$ and $\phi_{1}$ is the restriction of $\phi_{2}$ to the domain of $\phi_{1}$.

Proof.

The proof that $PI(\tilde{\Gamma})$ is an inverse semigroup that has the stated properties is a standard routine argument similar to the proof of the corresponding properties for $SIM(X)$. For example, the identity of $PI(\tilde{\Gamma})$ is clearly the identity automorphism of $\tilde{\Gamma}$. A partial isomorphism $\phi:\Delta_{1}\rightarrow\Delta_{2}$ is in the group of units of $PI(\tilde{\Gamma})$ if and only if $\Delta_{1}=\Delta_{2}=\tilde{\Gamma}$, so it follows that the group of units is $G(\tilde{\Gamma})$. ∎

If $\Delta$ is a connected subgraph of $\tilde{\Gamma}$ and $\tilde{v}$ is a vertex of $\Delta$, then we define $H(\Delta,\tilde{v})=f(L(\Delta,\tilde{v}))$. By Theorem 3.3 this is a closed inverse submonoid of $FIM(X)$ that is contained in $L(\Gamma,v)$ where $v=f(\tilde{v})$. In particular, if $\tilde{v}_{0}\in\Delta$, then $H(\Delta,\tilde{v}_{0})\leq H=H(\tilde{\Gamma},\tilde{v}_{0})$.

Lemma 4.2**.**

Let $\Delta_{1}$ and $\Delta_{2}$ be connected subgraphs of $\tilde{\Gamma}$ containing vertices $\tilde{v}_{1}$ and $\tilde{v}_{2}$ respectively. Then there is a partial isomorphism from $\Delta_{1}$ onto $\Delta_{2}$ that maps $\tilde{v}_{1}$ onto $\tilde{v}_{2}$ if and only if $H(\Delta_{1},\tilde{v}_{1})=H(\Delta_{2},\tilde{v}_{2})$.

Proof.

By Proposition 2.2 a partial isomorphism $\phi$ from $\Delta_{1}$ onto $\Delta_{2}$ that maps $\tilde{v}_{1}$ onto $\tilde{v}_{2}$ induces a label-preserving bijection from paths in $\Delta_{1}$ that start at $\tilde{v}_{1}$ to paths in $\Delta_{2}$ that start at $\tilde{v}_{2}$ and maps circuits to circuits. This bijection also maps $\sim_{i}$- equivalent paths to $\sim_{i}$- equivalent paths, so it induces an isomorphism from $L(\Delta_{1},\tilde{v}_{1})$ onto $L(\Delta_{2},\tilde{v}_{2})$. Also $f(\tilde{v}_{1})=f(\tilde{v}_{2})$. Denote this vertex of $\Gamma$ by $v$. Since $L(\Delta_{i},\tilde{v}_{i})\cong H(\Delta_{i},\tilde{v}_{i})$ by Theorem 3.3, this means that there is a labeled graph isomorphism from $H(\Delta_{1},\tilde{v}_{1})$ onto $H(\Delta_{2},\tilde{v}_{2})$ that fixes $v$. But this implies that $H(\Delta_{1},\tilde{v}_{1})=H(\Delta_{2},\tilde{v}_{2})$ since the labeling of edges in $\Gamma$ is consistent with an immersion into $B_{X}$.

Conversely, suppose that $H(\Delta_{1},\tilde{v}_{1})=H(\Delta_{2},\tilde{v}_{2})$. Let $\tilde{q_{1}}$ be a path in $\Delta_{1}$ starting at $\tilde{v}_{1}$ and let $q$ be its projection into $\Gamma$. Then $\tilde{q_{1}}\tilde{q}_{1}^{-1}\in L(\Delta_{1},\tilde{v}_{1})$, so $qq^{-1}\in H(\Delta_{1},\tilde{v}_{1})=H(\Delta_{2},\tilde{v}_{2})$, so $qq^{-1}$ lifts to a path $\tilde{q}_{2}\tilde{q}_{2}^{-1}$ with $\tilde{q}_{2}\tilde{q}_{2}^{-1}\in L(\Delta_{2},\tilde{v}_{2})$. Hence $q$ lifts to the path $\tilde{q}_{2}$ in $\Delta_{2}$ starting at $\tilde{v}_{2}$. Clearly ${\ell}(\tilde{q}_{1})={\ell}(\tilde{q}_{2})={\ell}(q)$. Dually, if $\tilde{q}_{2}$ is a path in $\Delta_{2}$ starting at $\tilde{v}_{2}$ then there is a corresponding path $\tilde{q}_{1}$ in $\Delta_{1}$ starting at $\tilde{q}_{1}$. The correspondence $\tilde{q}_{1}\leftrightarrow\tilde{q}_{2}$ is one-to-one by equality of labels on these paths. Since it also maps circuits to circuits, it induces an isomorphism from $\Delta_{1}$ onto $\Delta_{2}$ that maps $\tilde{v}_{1}$ onto $\tilde{v}_{2}$ by Proposition 2.2. ∎

If $v$ is a vertex in the connected graph $\Gamma$ and $H$ is a closed inverse submonoid of $L(\Gamma,v)$, then we denote by $N(H,v)$ the normalizer of $H$ in $L(\Gamma,v)$, that is $N(H,v)=\{p\in L(\Gamma,v):pHp^{-1},p^{-1}Hp\subseteq H\}$.

Lemma 4.3**.**

Let $f:\tilde{\Gamma}\rightarrow\Gamma$ be an immersion between connected graphs, $v_{0}$ a vertex of $\Gamma$, $\tilde{v}_{0}\in f^{-1}(v_{0})$ and $H=f(L(\tilde{\Gamma},\tilde{v}_{0}))$. Then

1. (a)

If there is a deck transformation that maps $\tilde{v}_{0}$ to $\tilde{v}_{1}$, then any path $\tilde{p}$ from $\tilde{v}_{0}$ to $\tilde{v}_{1}$ projects onto a path $p\in N(H,v_{0})$. 2. (b)

Each path $p\in N(H,v_{0})$ lifts to a path $\tilde{p}$ in $\tilde{\Gamma}$ starting at $\tilde{v}_{0}$. 3. (c)

Each path $p\in N(H,v_{0})$ determines a deck transformation ${\phi}_{p}$ of $\tilde{\Gamma}$ that maps $\tilde{v}_{0}$ to ${\omega}(\tilde{p})\in f^{-1}(v_{0})$. Furthermore, if $q$ is another element of $N(H,\tilde{v}_{0})$, then ${\phi}_{p}={\phi}_{q}$ iff ${\omega}(\tilde{p})={\omega}(\tilde{q})$.

Proof.

(a) Suppose there is a deck transformation that maps $\tilde{v}_{0}$ to $\tilde{v}_{1}$ and $\tilde{p}$ is a path from $\tilde{v}_{0}$ to $\tilde{v}_{1}$. Then $\tilde{p}L(\tilde{\Gamma},\tilde{v}_{1})\tilde{p}^{-1}\subseteq L(\tilde{\Gamma},\tilde{v}_{0})$ and $\tilde{p}^{-1}L(\tilde{\Gamma},\tilde{v}_{0})\tilde{p}\subseteq L(\tilde{\Gamma},\tilde{v}_{1})$. Since $f(L(\tilde{\Gamma},\tilde{v}_{1}))=f(L(\tilde{\Gamma},\tilde{v}_{0}))=H$ by Lemma 4.2, this implies that $pHp^{-1}\subseteq H$ and $p^{-1}Hp\subseteq H$, so $p\in N(H,v_{0})$.

(b) Let $p\in N(H,v_{0})$. Since $pp^{-1}\in H$ and every circuit in $H$ based at $v_{0}$ lifts to a circuit in $\tilde{\Gamma}$ based at $\tilde{v}_{0}$ it follows that $pp^{-1}$ lifts to a circuit which has the same label as $pp^{-1}$ based at $\tilde{v}_{0}$. By uniqueness of a path starting at $\tilde{v}_{0}$ with a given label, this circuit must be of the form $\tilde{p}\tilde{p}^{-1}$ for some lift $\tilde{p}$ of the path $p$ starting at $\tilde{v}_{0}$.

(c) Let $p$ and $\tilde{p}$ be as in part (b) above, denote $\omega(\tilde{p})$ by $\tilde{v}_{1}$ and let $H_{1}=f(L(\tilde{\Gamma},\tilde{v}_{1}))$. If $h\in H$, then $php^{-1}\in H$ since $p\in N(H,v_{0})$. This lifts to a circuit $\tilde{p}\tilde{h}_{1}\tilde{p}^{-1}$ based at $\tilde{v}_{0}$ since every circuit in $H$ lifts to a circuit in $\tilde{\Gamma}$ based at $\tilde{v}_{0}$. This forces $\tilde{h}_{1}$ to be a circuit in $\tilde{\Gamma}$ based at $\tilde{v}_{1}$ and with the same label as $h$, and so $h=f(\tilde{h}_{1})\in H_{1}$. Hence $H\subseteq H_{1}$.

Conversely, suppose that $h\in H_{1}$. Then $h$ lifts to a circuit $\tilde{h}_{1}$ in $\tilde{\Gamma}$ based at $\tilde{v}_{1}$. So $\tilde{p}\tilde{h}_{1}\tilde{p}^{-1}\in L(\tilde{\Gamma},\tilde{v}_{0})$, and so $php^{-1}\in H$. It follows that $p^{-1}php^{-1}p\in H$ since $p\in N(H,v_{0})$, so this lifts to a circuit of the form $\tilde{p}^{-1}\tilde{p}\tilde{h}\tilde{p}^{-1}\tilde{p}$ based at $\tilde{v}_{0}$ for some path $\tilde{h}$ with the same label as $h$. This path $\tilde{h}$ must be a circuit based at $\tilde{v}_{0}$ and so $f(\tilde{h})=h\in H$. Hence $H_{1}\subseteq H$ and so $H_{1}=H$. By Lemma 4.2 this implies that there is a deck transformation $\phi_{p}$ of $\tilde{\Gamma}$ that maps $\tilde{v}_{0}$ onto $\tilde{v}_{1}$. The last statement of part (c) of the lemma is immediate since automorphisms of an edge-labeled graph are determined by where they send a point. ∎

The following theorem provides an analogue for graph immersions Theorem 2.7.

Theorem 4.4**.**

Let $f:\tilde{\Gamma}\rightarrow\Gamma$ be an immersion of connected graphs with $\tilde{v}_{0}\in f^{-1}(v_{0})$ and $H=f(L(\tilde{\Gamma},\tilde{v}_{0}))$. Then $G(\tilde{\Gamma})\cong N(H,v_{0})/H$.

Proof.

Define $r:N(H,v_{0})\rightarrow G(\tilde{\Gamma})$ by $r(p)={\phi}_{p}$ in the notation of Lemma 4.3. If $p,q\in N(H,v_{0})$, then by Proposition 2.2 the deck transformation $\phi_{p}$ maps $\omega(\tilde{q})$ to $\omega(\tilde{q}_{1})$ where $\tilde{q}_{1}$ is a path starting at $\omega(\tilde{p})$ with $\ell(\tilde{q}_{1})=\ell(\tilde{q})$. So $\phi_{pq}(\tilde{v}_{0})=\omega(\tilde{p}\tilde{q}_{1})=\omega(\tilde{q}_{1})=\phi_{p}(\omega(\tilde{q}))=\phi_{p}(\phi_{q}(\tilde{v}_{0}))$, and so $\phi_{pq}=\phi_{p}\circ\phi_{q}$. Hence $r$ is a homomorphism. It is surjective by Lemma 4.3. Now $\tilde{\Gamma}$ is identified with the graph of right $\omega$-cosets of $H$ in $FIM(X)$ and under this identification, a vertex $\tilde{v}$ of $\tilde{\Gamma}$ is identified with the right $\omega$-coset $(Hp)^{\omega}$ where $p$ lifts to a path $\tilde{p}$ in $\tilde{\Gamma}$ starting at $\tilde{v}_{0}$ and ending at $\tilde{v}$. By Lemma 4.3, $\phi_{p}=\phi_{q}$ iff $\omega(\tilde{p})=\omega(\tilde{q})$, so the kernel of the map $r$ coincides with the equivalence relation $\rho_{H}$ that identifies $p$ and $q$ if $(Hp)^{\omega}=(Hq)^{\omega}$. It follows that $G(\tilde{\Gamma})\cong N(H,v_{0})/H$. ∎

5. Covers

In this section we characterize covers of graphs in terms of the concepts introduced earlier. A result related to part (a) of the following theorem was obtained by Meakin and Szakács [9] in the more general context of immersions between $2$-complexes.

Theorem 5.1**.**

Let $f:\tilde{\Gamma}\rightarrow\Gamma$ be an immersion of connected graphs. Choose basepoints $v_{0}\in\Gamma$ and $\tilde{v}_{0}\in\tilde{\Gamma}$ such that $\tilde{v}_{0}\in f^{-1}(v_{0})$ and let $H=f(L(\tilde{\Gamma},\tilde{v}_{0}))$. Then

1. (a)

$\tilde{\Gamma}$* is a cover of $\Gamma$ if and only if $H$ is a full inverse submonoid of $L(\Gamma,v_{0})$.* 2. (b)

$\tilde{\Gamma}$* is a normal cover of $\Gamma$ if and only if $N(H,v_{0})=L(\Gamma,v_{0})$ * 3. (c)

$\tilde{\Gamma}$* is the universal cover of $\Gamma$ if and only if $H$ consists of the idempotents in $L(\Gamma,v_{0})$, i.e. $H=\{p\in L(\Gamma,v_{0}):\ell(p)$ is a Dyck word $\}$.*

Proof.

(a) Suppose that $\tilde{\Gamma}$ is a cover of graphs. Then every path $p$ in $\Gamma$ that starts at $v_{0}$ lifts to a (unique) path $\tilde{p}$ starting at $\tilde{v}_{0}$ by Proposition 2.3. In particular, if $e$ is an idempotent of $L(\Gamma,v_{0})$, then $\ell(e)$ is a Dyck word since the only idempotents in $FIC(\tilde{\Gamma})$ are Dyck words, so $\ell(\tilde{e})$ is a path starting at $\tilde{v}_{0}$ whose label is a Dyck word. This forces $\tilde{e}$ to be a circuit at $\tilde{v}_{0}$, so $e=f(\tilde{e})\in H$. Hence $H$ is full in $L(\Gamma,v_{0})$. Conversely, suppose that $H$ is full in $L(\Gamma,v_{0})$. Let $v_{1}$ be any vertex in $\Gamma$ and $\tilde{v}_{1}$ any vertex in $f^{-1}(v_{1})$ and let $p$ be a path in $\Gamma$ starting at $v_{1}$. There is a path $\tilde{q}$ in $\tilde{\Gamma}$ from $\tilde{v}_{0}$ to $\tilde{v}_{1}$ and the projection of this path is a path $q$ in $\Gamma$ from $v_{0}$ to $v_{1}$. Then $qpp^{-1}q^{-1}$ is an idempotent in $L(\Gamma,v_{0})$ so it is in $H$ and hence it lifts to a path $\tilde{q}_{1}\tilde{p}_{1}\tilde{p}_{1}^{-1}\tilde{q}_{1}^{-1}$ in $\tilde{\Gamma}$ starting at $\tilde{v}_{0}$. Since $\ell(\tilde{q}_{1})=\ell(q)=\ell(\tilde{q})$ we must have $\tilde{q}_{1}=\tilde{q}$. Hence $\omega(\tilde{q}_{1})=\tilde{v}_{1}$ and so $\tilde{p}_{1}$ is a lift of $p$ that starts at $\tilde{v}_{1}$. Hence all paths in $\Gamma$ lift everywhere, so $\tilde{\Gamma}$ is a cover of $\Gamma$ by Proposition 2.3.

(b) Suppose that $\tilde{\Gamma}$ is a normal cover of $\Gamma$ and $v_{0}\in\Gamma$. Then for all vertices $\tilde{v}_{0},\tilde{v}_{1}\in f^{-1}(v_{0})$ there is a (unique) deck transformation $\phi$ mapping $\tilde{v}_{0}$ onto $\tilde{v}_{1}$. If $p\in L(\Gamma,v_{0})$ then $pp^{-1}$ is an idempotent in $L(\Gamma,v_{0})$ so $pp^{-1}\in H$ by part (a). Hence $pp^{-1}$ lifts to a path $\tilde{p}\tilde{p}^{-1}$ at $\tilde{v}_{0}$ and so $p$ lifts to the path $\tilde{p}$ from $\tilde{v}_{0}$ to some vertex $\tilde{v}_{1}$. Since there is a deck transformation $\phi$ that maps $\tilde{v}_{0}$ onto $\tilde{v}_{1}$, part (a) of Lemma 4.3 implies that $p\in N(H,v_{0})$. Hence $L(\Gamma,v_{0})\subseteq N(H,v_{0})$ and so $L(\Gamma,v_{0})=N(H,v_{0})$.

Conversely, suppose that $L(\Gamma,v_{0})=N(H,v_{0})$ and $e$ is an idempotent in $L(\Gamma,v_{0})$. Then $e\in N(H,v_{0})$. But $H$ is full in $N(H,v_{0})$ by Proposition 3.4, so $e\in H$. Hence $\tilde{\Gamma}$ is a cover of $\Gamma$ by part (a). Now let $\tilde{v}_{0},\tilde{v}_{1}\in f^{-1}(v_{0})$. There is some path $\tilde{r}$ from $\tilde{v}_{0}$ to $\tilde{v}_{1}$ that projects to a path $r\in L(\Gamma,v_{0})=N(\Gamma,v_{0})$. So by Lemma 4.3 there is a deck transformation that maps $\tilde{v}_{0}$ to $\tilde{v}_{1}$, and so $\tilde{\Gamma}$ is a normal cover of $\Gamma$.

(c) Suppose that $\tilde{\Gamma}$ is the universal cover of $\Gamma$. Then $\tilde{\Gamma}$ is a tree, so the label of any circuit $\tilde{p}$ in $\tilde{\Gamma}$ based at any point is a Dyck word. Such circuits project onto circuits in $\Gamma$ whose label is also a Dyck word, so $H$ consists just of idempotents in $L(\Gamma,v_{0})$. On the other hand, if $e$ is an idempotent in $L(\Gamma,v_{0})$ then $\ell(e)$ is a Dyck word (since any idempotent in $FIC(\Gamma)$ is a path whose label is a Dyck word). Since $e$ lifts to a path $\tilde{e}$ based at $\tilde{v}_{0}$ and since $\ell(\tilde{e})$ is a Dyck word, $\tilde{e}$ is a circuit based at $\tilde{v}_{0}$, so $f(\tilde{e})=e\in H$. Hence $H$ consists exactly of all the Dyck words in $L(\Gamma,v_{0})$. Conversely, if $H$ consists of these idempotents, then $H$ is full in $L(\Gamma,v_{0})$ so $\tilde{\Gamma}$ is a cover of $\Gamma$ by part (a). By definition of $H$, any circuit in $\tilde{\Gamma}$ based at $\tilde{v}_{0}$ projects onto a circuit in $H$ based at $v_{0}$, so its label must be a Dyck word. It follows that $\tilde{\Gamma}$ is a tree. Hence $\tilde{\Gamma}$ is the universal cover of $\Gamma$. ∎

6. Actions of groups on graphs

By an action of a group $G$ on a graph $\Gamma$ we mean a homomorphism $\phi:G\rightarrow Aut(\Gamma)$ of $G$ into the group of (labeled graph) automorphisms of $\Gamma$. Here we adopt the convention that the action is a left action and denote the image under $\phi(g)$ of the vertex $v$ [or edge $e$] by $g.v$ [resp. $g.e$]. If $f:\tilde{\Gamma}\rightarrow\Gamma$ is an immersion between connected graphs then there is an obvious action of the group $G(\tilde{\Gamma})$ on $\tilde{\Gamma}$. The following fact is well-known for covers of connected graphs, but also holds for immersions (with essentially the same proof).

Proposition 6.1**.**

If $f:\tilde{\Gamma}\rightarrow\Gamma$ is an immersion of connected graphs with group $G=G(\tilde{\Gamma})$ of deck transformations, then the quotient map $g:\tilde{\Gamma}\rightarrow\tilde{\Gamma}/{G}$ is a normal covering and $G$ is the group of deck transformations of this cover.

Proof.

If $\gamma$ is a deck transformation of the immersion $f$ then by Proposition 2.1, $\gamma$ is uniquely determined by the image of any vertex in $\Gamma$, and it follows that if $v$ is a vertex and $e$ is a edge of $\Gamma$ and $\gamma$ is not the identity automorphism of $\Gamma$, then $\gamma(v)\neq v$ and $\gamma(e)\neq e$. From this it follows that the action of $G$ on $\tilde{\Gamma}$ satisfies condition (*) on page 72 of Hatcher [3] and so the result follows by Proposition 1.40 of [3]. ∎

**Example ** The graph $\tilde{\Gamma}$ in Figure 3 has four vertices $v_{1},v_{2},v_{3}$ and $v_{4}$ and six positively labeled edges. It immerses in the obvious way into the bouquet $B_{\{a,b\}}$ of two circles via the map $f$ that preserves edge labels.

aaaaaabbbgw_{1}$$w_{2}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$\Gamma$$\tilde{\Gamma}

Figure 3

Let $e_{1}$ be the directed edge from $v_{1}$ to $v_{4}$ with label $a$; $e_{2}$ the directed edge from $v_{2}$ to $v_{3}$ with label $a$; $e_{3}$ the directed edge from $v_{3}$ to $v_{2}$ with label $a$; $e_{4}$ the directed edge from $v_{4}$ to $v_{1}$ with label $a$; $e_{5}$ the directed edge from $v_{1}$ to $v_{2}$ with label $b$; and $e_{6}$ the directed edge from $v_{4}$ to $v_{3}$ with label $b$.

The stabilizer of the vertex $v_{1}$ under the action by $FIM(a,b)$ is the closed inverse submonoid $H$ of $FIM(a,b)$ generated by the elements $ba^{2}b^{-1},aba^{-1}b^{-1}$ and $bab^{-1}a$. The graph $\tilde{\Gamma}$ is the graph of right $\omega$-cosets of $H$ in $FIM(a,b)$. The distinct right $\omega$-cosets of $H$ are of course $H,(Hb)^{\omega},(Hba)^{\omega}$ and $(Ha)^{\omega}$, and they may be identified with the four vertices $v_{1},v_{2},v_{3}$ and $v_{4}$ respectively.

The map that interchanges $e_{1}$ and $e_{4}$, interchanges $e_{2}$ and $e_{3}$, and interchanges $e_{5}$ and $e_{6}$ defines a deck transformation $\gamma$ of the immersion $f$ that interchanges $v_{1}$ with $v_{4}$ and $v_{2}$ with $v_{3}$. This is the only non-trivial deck transformation, so the group $G=G(\tilde{\Gamma})$ is isomorphic to the cyclic group ${\mathbb{Z}}_{2}$ of order $2$. Note that the only right $\omega$-coset of $H$ that lies in $N(H)$ is $(Ha)^{\omega}$, so $N(H)/H=\{H,(Ha)^{\omega}\}$ is isomorphic to ${\mathbb{Z}}_{2}$ in accord with Theorem 4.4.

The group $G={\mathbb{Z}}_{2}$ acts on $\tilde{\Gamma}$ in the obvious way and the quotient graph $\Gamma=\tilde{\Gamma}/G$ is the graph with two vertices $w_{1},w_{2}$ and one positively oriented edge from $w_{1}$ to $w_{2}$ with label $b$, as depicted in Figure 3. Clearly the map $g$ from $\tilde{\Gamma}$ to $\Gamma$ that preserves edge labels is a normal cover of $\Gamma=\tilde{\Gamma}/G$, in accord with Proposition 6.1.

7. Extending immersions to covers

Theorem 7.1**.**

Let $f:\tilde{\Gamma}\rightarrow\Gamma$ be an immersion of connected graphs. Then there is a graph cover $g:\tilde{\Delta}\rightarrow\Gamma$ such that

(a) $\tilde{\Gamma}$ is a subgraph of $\tilde{\Delta}$ and $f$ is the restriction of $g$ to $\tilde{\Gamma}$; and

(b) any deck transformation of $\tilde{\Gamma}$ is the restriction of some deck transformation of $\tilde{\Delta}$.

Proof.

(a) If $f:\tilde{\Gamma}\rightarrow\Gamma$ is a cover of graphs there is nothing to prove, so assume that this is not the case. Let the edges of $\Gamma$ and $\tilde{\Gamma}$ be labeled over a set $X\cup X^{-1}$ consistent with an immersion into $B_{X}$ as usual. Then there is a vertex $\tilde{v}$ of $\tilde{\Gamma}$ for which there is an edge $e$ in $\Gamma$ with $f(\tilde{v})=\alpha(e)$ such that $e$ does not lift to any edge in $\tilde{\Gamma}$ starting at $\tilde{v}$. If $\ell(e)=x\in X\cup X^{-1}$ then there is no edge with label $x$ starting at $\tilde{v}$. We refer to such a vertex as an incomplete vertex of $\tilde{\Gamma}$ and say that $\tilde{v}$ is missing an edge labeled by $x$. Enlarge the graph $\tilde{\Gamma}$ by adding a new vertex $\tilde{v}_{x}$ and a new edge $\tilde{e}_{x}$ from $\tilde{v}$ to $\tilde{v}_{x}$ for each incomplete vertex $\tilde{v}$ of $\tilde{\Gamma}$ that is missing an edge labeled by $x$. Since distinct edges starting at $f(\tilde{v})$ have distinct labels, the new vertices and edges that we added to $\tilde{\Gamma}$ are all distinct. Clearly the graph $\tilde{\Delta}_{1}$ immerses into $\Gamma$ via the map that preserves edge labeling. Then apply the same process to $\Delta_{1}$, adding new edges as necessary at any incomplete vertices, to form the graph $\Delta_{2}$. Continue in this fashion to build a sequence of graphs $\tilde{\Gamma}\subseteq\tilde{\Delta}_{1}\subseteq\tilde{\Delta}_{2}\subseteq...$ by adding new vertices and edges to the previous graph at any incomplete vertices. Let $\tilde{\Delta}$ be the union of the graphs $\tilde{\Delta}_{i}$ as $i$ ranges from $1$ to $\infty$. The graph $\tilde{\Delta}$ is obtained from $\tilde{\Gamma}$ by adding (possibly infinite) trees to incomplete vertices of $\tilde{\Gamma}$. Then the map $g:\tilde{\Delta}\rightarrow\Gamma$ that extends $f$ and maps paths in $\tilde{\Delta}\setminus\tilde{\Gamma}$ to their obvious images in $\Gamma$ is a covering map since $\tilde{\Delta}$ has no incomplete vertices.

(b) Suppose now that $\gamma$ is a deck transformation of the immersion $f:\tilde{\Gamma}\rightarrow\Gamma$ that takes a vertex $\tilde{v}$ to a vertex $\tilde{v}_{1}$. Any path $\tilde{p}$ in $\tilde{\Delta}$ starting at $\tilde{v}$ factors in the form $\tilde{p}=\tilde{p}_{1}\tilde{q}_{1}\tilde{p}_{2}\tilde{q}_{2}...\tilde{p}_{n}\tilde{q}_{n}$ where $\tilde{p}_{1}\tilde{p}_{2}...\tilde{p}_{n}$ is a path in $\tilde{\Gamma}$ and the $\tilde{q}_{i}$ are circuits in the forest $\tilde{\Delta}\setminus\tilde{\Gamma}$ based at $\omega(\tilde{p}_{i})$ for $i=1,...,n-1$ and $\tilde{q}_{n}$ is a path in $\tilde{\Delta}\setminus\tilde{\Gamma}$ starting at $\omega(\tilde{p}_{n})$. By Proposition 2.2, there is a path $\tilde{p}_{1}^{\prime}$ in $\tilde{\Gamma}$ starting at $\tilde{v}_{1}$ with $\ell(\tilde{p}_{1}^{\prime})=\ell(\tilde{p}_{1})$. Also by the same proposition, there is an edge $\tilde{e}$ in $\tilde{\Gamma}$ starting at $\omega(\tilde{p}_{1})$ with label $x$ if and only if there is an edge $\tilde{e}^{\prime}$ in $\tilde{\Gamma}$ starting at $\omega(\tilde{p}^{\prime})$ with the same label. It follows that there is a path of the form $\tilde{q}_{1}^{\prime}$ in $\tilde{\Delta}\setminus\tilde{\Gamma}$ with $\ell(\tilde{q}_{1}^{\prime})=\ell(\tilde{q}_{1})$, $\alpha(\tilde{q}_{1}^{\prime})=\omega(\tilde{p}_{1}^{\prime})$ and $\tilde{q}_{i}^{\prime}$ is a circuit if and only if $\tilde{q}_{i}$ is a circuit. Continuing in this fashion, we see that there is a path $\tilde{p}^{\prime}=\tilde{p}_{1}^{\prime}\tilde{q}_{1}^{\prime}\tilde{p}_{2}^{\prime}\tilde{q}_{2}^{\prime}...\tilde{p}_{n}^{\prime}\tilde{q}_{n}^{\prime}$ with $\ell(\tilde{p}^{\prime})=\ell(\tilde{p})$. Furthermore, $\tilde{p}^{\prime}$ is a circuit if and only if $\tilde{p}$ is a circuit since $\omega(\tilde{p}^{\prime})=\gamma(\omega(\tilde{p}))$. Hence by Proposition 2.2 there is a deck transformation $\tilde{\gamma}$ of $\tilde{\Delta}$ that takes $\tilde{v}$ to $\tilde{v}_{1}$. The bijection $\tilde{\gamma}$ extends $\gamma$ and maps a path in $\tilde{\Delta}\setminus\tilde{\Gamma}$ starting at an incomplete vertex $\tilde{w}$ to the path with the same label starting at $\gamma(\tilde{w})$. ∎

We remark that it is not clear that it is possible to extend finite immersions between graphs to finite covers in such a way that deck transformations of the immersion are restrictions of deck transformations of the cover.

Acknowledgement. We are grateful to Nora Szakács for pointing out an error in an earlier version of this paper.