\plparsep
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On algebraic extensions and decomposition
of homomorphisms of free groups
Noam Kolodner
Abstract
We give a counterexample to a conjecture by Miasnikov, Ventura and
Weil, stating that an extension of free groups is algebraic if and
only if the corresponding morphism of their core graphs is onto, for
every basis of the ambient group. In the course of the proof we present
a partition of the set of homomorphisms between free groups which
is of independent interest.
1 Introduction
The purpose of this paper is to develop new machinery for the study
of morphisms between core graphs associated with free groups (see
definitions in §2). For this purpose, we construct
a category which enriches that of free groups and enables to study
surjectivity with respect to arbitrary bases. The main application
we give is a counterexample to a conjecture by Miasnikov, Ventura,
and Weil [MR2395796] which was revised by Parzanchevski and
Puder [MR3211804].
Conjecture 1.1** ([MR2395796]).**
Let FY be a free group on the set Y, and
let H≤K≤FY be subgroups. If the morphism between the
associated core-graphs ΓX(H)→ΓX(K)
is surjective for every basis X of FY, then K is an algebraic
extension of H.
The converse is indeed true: if H≤K is an algebraic extension,
then ΓB(H)→ΓB(K) is indeed
onto for any basis B. In [MR3211804], Parzanchevski and Puder
constructed a counterexample for the conjecture in F2, the free
group on two generators. Their proof relies heavily on idiosyncrasies
of the automorphism group of F2, and they conjectured that these
are the only obstructions, revising the conjecture in two ways:
Conjecture 1.2** ([MR3211804]).**
Let H≤K≤FY. If for every free extension
F′ of FY and for every basis X of F′ the morphism ΓX(H)→ΓX(K)
is onto then K is algebraic over H.
Conjecture 1.3** ([MR3211804]).**
The original conjecture holds for H≤K≤FY
such that ∣Y∣>2.
In this paper, we show that both revised forms of the conjecture are
false. In fact, we prove a stronger statement:
Theorem 1.4**.**
The extension ⟨bbaba−1⟩<⟨b,aba−1⟩
is not algebraic, but for every morphism φ:F{a,b}→FX
with X arbitrary and φ(a),φ(b)=1, the graph morphism
ΓX(φ(⟨bbaba−1⟩))→ΓX(φ(⟨b,aba−1⟩))
is surjective.
Our result naturally raises the question what is the algebraic interpretation
(if one exists at all) of the property of being “onto in all bases”,
as the one suggested by the conjectures above turns out to be false.
In addition, the methods which we develop for the analysis of our
counterexample are of independent interest: The first idea is extending
the scope of inspection from the category of automorphisms to a larger
category of homomorphisms called FGR (“Free Groups with Restrictions”),
which is defined in §3. In the new category
we present a recursive decomposition of morphisms, which in turn enables
ones to verify whether the desired property of graphs holds for all
FGR homomorphisms. A second point of interest is the observation of
a property of the word bbaba−1 which ensures that the recursive
process ends. This property is defined in §6,
and leads to further questions and applications, for example to solving
some forms of equations in free groups.
Acknowledgement*.*
I am grateful to Ori Parzanchevski for his guidance and for carefully
reading and commenting on a draft of this paper. I would also like
to thank the referee for the helpful comments. This research was supported
by ISF grant 1031/17 of Ori Parzanchevski.
2 Preliminaries
We present the definitions we use in the paper, merging the language
of [MR1882114] and [MR695906].
Definition 2.1** (Graphs).**
We use graphs in the sense of Serre [MR0476875]: A graph
Γ is a set V(Γ) of vertices and a set E(Γ)
of edges with a function ι:E(Γ)→V(Γ)
called the initial vertex map and an involution \bigbox:E(Γ)→E(Γ)
with e=e and e=e. A graph
morphism f:Γ→Δ is a pair of set functions fE:E(Γ)→E(Δ)
and fV:V(Γ)→V(Δ) that
commute with the structure functions. A *path *in Γ is
a finite sequence e1,…,en∈E(Γ) with ι(ek)=ι(ek+1)
for every 1≤k<n. The path is closed if ι(en)=ι(e1),
and reduced if ek+1=ek for all k.
All the graphs in the paper are assumed to be connected unless specified
otherwise, namely, for every e,f∈E(Γ) there
is a path e,e1,…,en,f.
Definition 2.2** (Labeled graphs).**
Let X be a set and let X−1 be the set of its formal inverses.
We define RX to be the graph with E(RX)=X∪X−1,
V(RX)={∗}, x=x−1
and ι(x)=∗. An *X-labeled graph *is a graph
Γ together with a graph morphism l:Γ→RX.
A morphism of X-labeled graphs Γ and Δ is a graph
morphism f:Γ→Δ that commutes with the label functions.
Let P(Γ) be the set of all the paths
in Γ, let FX be the free group on X and let P=e1,…,en
be a path. The edge part of the label function lE:E(Γ)→E(RX)
can be extended to a function l:P(Γ)→FX
by the rule l(P)=l(e1)…(en).
A *pointed *X-labeled graph is an X-labeled graph that
has a distinguished vertex called the base point. A morphism of a
pointed labeled graph sends the base point to the base point. This
constitutes a category called X-Grph. For a pointed X-labeled
graph Γ we define π1(Γ) to be
[TABLE]
Definition 2.3**.**
Let FX be the free group on the set X. We define a category
Sub(FX), whose objects are subgroups H≤FX
and there is a unique morphism in Hom(H,K) iff H≤K.
It is easy to verify that π1 is a functor from X-Grph
to Sub(FX).
Note: The functor π1 defined above is not the fundamental
group of Γ as a topological space. Rather, if one views Γ
and RX as topological spaces and l as a continuous function,
then π1 is the image of the fundamental group of Γ
in that of RX under the group homomorphism induced by l.
Definition 2.4** (Folding).**
A labeled graph Γ is *folded *if l(e)=l(f)
holds for every two edges e,f∈E(Γ) with ι(f)=ι(e).
We notice that there is at most one morphism between two pointed folded
labeled graphs. If Γ is not folded, there exist e,f∈E(Γ)
such that ι(e)=ι(f) and l(e)=l(f);
Let Γ′ be the graph obtained by identifying the vertex ι(e)
with ι(f), the edges e with f and
e with f. We say Γ′ is the result
of folding e and f. The label function l factors through
Γ′, yielding a label function l′ on Γ′, and we
notice that π1(Γ)=π1(Γ′).
Definition 2.5** (Core graph).**
A core graph Γ is a labeled, folded, pointed graph
such that every edge in Γ belongs in a closed reduced path
around the base point. Is case of finite graphs this is equivalent
to every v∈V(Γ) having deg(v):=∣ι−1(v)∣>1
except the base point which can have any degree.
Definition 2.6**.**
Let X-CGrph be the category of connected, pointed, folded,
X-labeled core graphs. Define a functor ΓX:Sub(FX)→X-CGrph
that associates to the subgroup H≤FX a graph ΓX(H)
(which is unique up to a unique isomorphism) such that π1(ΓX(H))=H.
Fact 2.7** ([MR695906, MR1882114]).**
The functors π1 and Γ define an equivalence between
the categories X-CGrph and Sub(FX).
The correspondence between the categories of X-CGrph and
Sub(FX) follows from the theory of cover spaces.
Let us sketch a proof. We regard RX as a topological space and
look at the category of connected pointed cover spaces of RX.
This category is equivalent to Sub(FX), following
from the fact that RX has a universal cover. Let Γ be
a connected folded X-labeled core graph, viewed as a topological
space and l as a continuous function. There is a unique way up
to cover isomorphism to extend Γ to a cover of RX. There
is also a unique way to associate a core graph to a cover space of
RX. This gives us an equivalence between the category of connected
pointed cover spaces of RX and pointed connected folded X-labeled
core graphs.
Definition 2.8**.**
By uniqueness of the core graph of a subgroup we can define a functor
\text{Core}\colon\text{X-Grph}\to X\text{-CGrph} that associates
to a graph Γ a core graph such that π1(Core(Γ))=π1(Γ).
Definition 2.9**.**
Let Γ be a graph with a vertex v of degree one which is
not the base-point. For e=ι−1(v), let Γ′
be the graph with V(Γ′)=V(Γ)\{v}
and E(Γ′)=E(Γ)\{e,e}.
We say that Γ′ is the result of trimming e from
Γ, and we notice that π1(Γ)=π1(Γ′).
Remark 2.10**.**
For a finite graph Γ, after both trimming and folding ∣E(Γ′)∣<∣E(Γ)∣.
If no foldings or trimmings are possible then Γ is a core
graph. This means that after preforming a finite amount of trimmings
and foldings we arrive at Core(Γ). It follows
from the uniqueness of Core(Γ) that the order
in which one performs the trimmings and foldings does not matter.
Definition 2.11** (Algebraic extension).**
Let H≤K≤FX be subgroups. The subgroup K is said
to be an *algebraic extension *of H if there exists no proper
free factor J of K with H≤J<K.
Conjecture 1.1 is the converse to the following:
Fact 2.12** ([MR2395796]).**
If H≤K≤FX and H≤K is
an algebraic extension then ΓY(H)→ΓY(K)
is onto for every basis Y of FX.
Another motivation for this conjecture is the fact that in a specific
case it is true
Fact 2.13**.**
Let Y be a finite set then FY is
algebraic over H iff ΓX(H)→ΓX(FY)
is onto for every basis X of FY (To see why this is true
notice that ΓX(FY) is a bouquet of circles
for every X).
When we combine Facts 2.12 and 2.13
we get a statement about intermediate subgroups which is another motivation
for the conjecture: Let H≤K≤FY then K is algebraic
over H iff for every L with K≤L≤FY and every basis
X of L the morphism ΓX(H)→ΓX(K)
is onto.
3 The category FGR
Definition 3.1** (Whitehead graph).**
A 2-path in a graph Γ is a pair (e,f)∈E(Γ)×E(Γ)
with ι(f)=ι(e) and f=e.
If Γ is X-labeled, the set
[TABLE]
forms the set of edges of a combinatorial (undirected) graph whose
vertices are X∪X−1, called the *Whitehead graph *of
Γ. If w∈FX is a cyclically reduced word, the Whitehead
graph of w as defined in [MR1575455, MR1714852] and the Whitehead
graph of Γ(⟨w⟩) defined
here coincide. Let WX=W(RX) be the set of edges
of the Whitehead graph of RX, which we call the full Whitehead
graph. Let x,y∈X∪X−1 and let {x,y}∈WX
be an edge. We denote x.y={x,y} (this is similar
to the notation in [MR1812024]).
Definition 3.2**.**
A homomorphism φ:FY→FX is non-degenerate
if φ(y)=1 for every y∈Y.
Definition 3.3**.**
Let w∈FX be a reduced word of length n. Define Γw
to be the X-labeled graph with V(Γw)={1,…,n+1}
forming a path P labeled by l(P)=w. Notice that Γw≅Γw−1.
Definition 3.4**.**
Let φ:FY→FX be a non-degenerate homomorphism.
We define a functor Fφ from Y-labeled graphs
to X-labeled graphs by sending y-labeled edges to φ(y)-labeled
paths. Formally, let Δ be a Y-labeled graph and let E0={e∈E(Δ)∣l(e)∈Y}
be an orientation of Δ, namely, E(Δ)=E0⊔{e∣e∈E0}.
For every e∈E(Δ) let ne∈N be
the length of the word φ(l(e))∈FX
plus one. We consider V(Δ) as a graph without edges,
take the disjoint union of graphs ⨆e∈E0Γφ(l(e))⊔V(Δ)
and for every e∈E0 glue 1∈V(Γφ(l(e)))
to ι(e)∈V(Δ), and ne∈V(Γφ(l(e)))
to ι(e)∈V(Δ). As for
functoriality, if f:Δ→Ξ is a morphism of Y-labeled
graphs, Fφf is defined as follows: every edge
in Fφ(Δ) belongs to a path Fφ(e)
for some e∈E(Δ), and we define (Fφf)(Fφ(e))=Fφ(f(e)).
The motivation for Fφ is topological: thinking
of Δ,RY,RX as topological spaces and of l:Δ→RY,φ:RY→RX
as continuous functions, we would like to think of Δ as an
X-labeled graph with the label function φ∘l. The
problem is that φ∘l does not send edges to edges, and
we mend this by splitting edges in Δ to paths representing
their images in RX.
Remark 3.5**.**
For H≤FY we notice that Core(FφΓY(H))=ΓX(φ(H)).
Definition 3.6** (Stencil).**
Let Γ be an Y-labeled graph, and φ:FY→FX
a non-degenerate homomorphism. We say that the pair (φ,Γ)
is a stencil iff Fφ(Γ)
is a folded graph. Notice that if Γ is not folded, then Fφ(Γ)
is not folded for any φ.
Definition 3.7**.**
Let τ:FX\{1}→X∪X−1
be the function returning the last letter of a reduced word. For reduced
words u,v in a free group, we write u⋅v to indicate that
there is no cancellation in their concatenation, namely τ(u)=τ(v−1).
We leave the proof of Lemmas 3.8 and 3.10
as easy exercises for the reader.
Lemma 3.8**.**
Let Γ be an Y-labeled graph and φ:FY→FX
a non-degenerate homomorphism. Then (φ,Γ)
is a stencil iff Γ is folded and for every x.y∈W(Γ)
we have φ(x)⋅φ(y)−1 (i.e. τ(φ(x))=τ(φ(y))).
Proposition 3.9**.**
Let φ:FZ→FY and ψ:FY→FX
be non-degenerate homomorphisms. The equality Fψ∘φ=Fψ∘Fφ
holds iff (ψ,Γφ(x)) is a stencil for every x∈Z.
Proof.
As Fφ is defined by replacing edges by paths
and gluing, it is enough to consider the graph Γx with
a single edge labeled x∈Z. Both Fψ∘φ(Γx)
and Fψ∘Fφ(Γx)
are paths whose labels equal ψ(φ(x)),
but Fψ∘φ(Γx) is always
folded whereas Fψ∘Fφ(Γx)
may not be. In fact, Fφ(Γx)=Γφ(x)
and so by definition Fψ∘Fφ(Γx)
is folded iff (ψ,Γφ(x)) is a stencil.
∎
Lemma 3.10**.**
Let φ and ψ be homomorphism as
in Proposition 3.9 (such that Fψ∘φ=Fψ∘Fφ)
and let x∈Z. Then τψ(φ(x))=τψ(τφ(x)).
Definition 3.11** (FGR).**
The objects of the category Free Groups with Restrictions
(FGR) are pairs (Y,N) where Y is a set
of “generators” and N⊆WX a set of “restrictions”.
A morphism φ∈Hom((Y,N),(X,M))
is a group homomorphism φ:FY→FX with
the following properties:
- (i)
For every x∈Y, φ(x)=1 (φ is
non-degenerate).
2. (ii)
For every x∈Y, W(Γφ(x))⊆M.
3. (iii)
For every x.y∈N, φ(x)⋅φ(y)−1 (i.e. τ(φ(x))=τ(φ(y))).
4. (iv)
For every x.y∈N, τ(φ(x)).τ(φ(y))∈M.111Technically, (iv) implies (iii), as M⊆WY and x.x∈/WY.
A main motivation for this definition is the second part of the next
Proposition.
Proposition 3.12**.**
-
FGR* is a category, i.e. composition of morphisms is
a morphism.*
2. 2.
Any two composable FGR morphisms φ,ψ satisfy Fψ∘φ=Fψ∘Fφ.
Proof.
Let (X1,N1)⟶φ(X2,N2)⟶ψ(X3,N3).
For any x∈X1 we have W(Γφ(x))⊆N2
since φ satisfies (ii). Since ψ satisfies (iii), (ψ,Γφ(x))
is a stencil, so that Fψ∘φ=Fψ∘Fφ
by Proposition 3.9. This also implies lenψ(φ(x))≥lenφ(x)≥1,
so that ψ∘φ satisfies (i). As Γψ∘φ(x)=Fψ(Γφ(x)),
we have
[TABLE]
The part in [ ] is contained in N3 since ψ
satisfies (ii), and the rest is contained in N3 since φ
satisfies (ii) and ψ satisfies (iv), hence ψ∘φ
satisfies (ii). Finally, whenever x.y∈N1, it follows from
(iv) for φ that τφ(x).τφ(y)∈N2,
and from (iv) for ψ that τψ(τφ(x)).τψ(τφ(y))∈N3.
Using Lemma 3.10 we see that τψ(φ(x)).τψ(φ(y))∈N3,
hence ψ∘φ satisfies (iv), and thus also (iii).
∎
Lemma 3.13**.**
An FGR morphism φ:(Y,N)→(X,M)
is an FGR-isomorphism iff φ∣Y∪Y−1:Y∪Y−1→X∪X−1
is a bijection that commutes with inversion and M={φ(x).φ(y)∣x.y∈N}.
Proof.
We show that if φ:(Y,N)→(X,M)
is an FGR-isomorphism then φ∣Y∪Y−1:Y∪Y−1→X∪X−1,
and the rest is immediate. Let φ and ψ be composable
FGR morphisms. As (ψ,Γφ(x))
is a stencil, for every x∈Y we have len(ψ∘φ(x))≥len(φ(x)).
Let x∈Y and let φ be an isomorphism and ψ its
inverse. So len(φ(x))≤len(ψ∘φ(x))=len(x)=1,
and on the other hand φ(x)=1 implies len(φ(x))=1,
namely φ(x)∈X∪X−1.
∎
3.1 Partition of Hom in FGR
Let X be a countably infinite set, Y a finite set and (Y,NY)
an FGR object. We present a recursive partition of Hom((Y,NY),(X,WX)).
The ultimate goal of this partition is decomposing free group morphisms
FY→FX, and we notice that Hom((Y,∅),(X,WX))
are all the non-degenerate homomorphisms in Hom(FY,FX).
In order to perform this recursive process we need to consider morphisms
with a general restriction set NY in the domain, but it is not
necessary to consider a general NX in place of WX (and
in fact, the method presented does not work for a general restriction
set NX).
Let φ∈Hom((Y,NY),(X,WX)). If x.y∈NY then
φ(x)⋅φ(y−1) by definition. Let x.y∈WY\NY.
There are five possible types of cancellation in the product φ(x)φ(y−1).
Denote u=φ(x), v=φ(y), and
let t be the maximal subword canceled in uv−1, so that u=u0⋅t,
v=v0⋅t, and uv−1=u0⋅v0−1. The five
types of cancellation are:
-
No cancellation, namely t=1.
2. 2.
u and v−1 do not absorb one another, namely t,u0,v0=1
and uv−1=u0⋅v0−1.
3. 3.
u absorbs v−1, namely t=1, v0=1,u0=1 and
uv−1=u0.
4. 4.
v−1 absorbs u, namely t=1, v0=1,u0=1 and
uv−1=v0−1.
5. 5.
u and v−1 cancel each other out, namely u=v.
If y=x−1 then only the first two types can occur, and if x−1.y−1∈NY
only the first four. For φ∈Hom((Y,NY),(X,WX))
and x.y∈WY\NY such that φ(x)φ(y−1)
is of cancellation type i, we define an FGR object (Ui,NUi)
and a so-called folding morphism ψx.yi∈Hom((Y,NY),(Ui,NUi)),
such that φ factors through ψx.yi. Namely,
there is an FGR morphism φ′∈Hom((Ui,NUi),(X,WX))
with φ=φ′∘ψx.yi. We describe Ui,NUi,ψx.yi,φ′
for each type:
-
U1=Y, ψx.y1=Id, NU1=NY∪{x.y},
and φ′=φ.
2. 2.
There are two cases here, depending on whether x=y−1 or not:
x=y−1: U2=Y⊔{s}
and NU2={τψx.y2(r).τψx.y2(z)∣r.z∈NY}∪{x.s−1,y.s−1,x.y}
where
[TABLE]
x=y−1: U2=Y⊔{s}
and NU2={τψx.y2(r).τψx.y2(z)∣r.z∈NY}∪{x.s−1,y.s−1,x.y}
where
[TABLE]
3. 3.
U3=Y and NU3={τψx.y3(r).τψx.y3(z)∣r.z∈NY}∪{x.y−1}
where
[TABLE]
4. 4.
U4=Y and NU4={τψx.y4(r).τψx.y4(z)∣r.z∈NY}∪{x−1.y}
where
[TABLE]
5. 5.
U5=Y−{y} and NU5={τψx.y5(r).τψx.y5(z)∣r.z∈NY}
where
[TABLE]
With x.y fixed, let ψx.yi∗:Hom((Ui,NUi),(X,WX))→Hom((Y,NY),(X,WX))
be the induced function φ′↦φ′∘ψx.yi.
We obtain that Hom((Y,NY),(X,WX))
can be presented as a disjoint union
[TABLE]
For every (Ui,NUi), if NUi=WUi we continue
recursively, choosing an edge x.y∈WUi\NUi
and partitioning Hom((Ui,NUi),(X,WX))
accordingly. We now show that no matter which edges are chosen, this
process terminates:
Theorem 3.14**.**
Any φ∈Hom((Y,NY),(X,WX))
decomposes as φ=φ′∘ψk∘⋯∘ψ1
such that
-
The morphisms ψj:Uj−1→Uj (1≤j≤k,
U0=Y) are folding morphisms.
2. 2.
NUk=WUk,* and in particular φ′(x)⋅φ′(y)−1
for x=y∈Uk∪Uk−1.*
Proof.
Let *O *be the collection of all FGR morphisms from
some (Y,NY) with Y finite into (X,WX). Define a height
function h:O→N×N by h\left(\varphi\right)=\big{(}\sum_{y\in Y}\text{len}\left(\varphi\left(y\right)\right),\left|W_{Y}\backslash N_{Y}\right|\big{)},
and consider N×N with the lexicographic
order. It is straightforward to verify that h(φ′)<h(φ)
for every folding morphism ψx.yi and φ=φ′∘ψx.yi.
Thus, the decomposition process ends in a finite amount of steps.
∎
Proposition 3.15** (The triangle rule).**
Let (Y,NY) be an FGR object, and x.y∈NY
and z∈Y∪Y−1 be such that z.y,z.x∈WY\NY.
Then
[TABLE]
(not a disjoint union), where NY1=NY∪{z.y}
and NY2=NY∪{z.x}.
Proof.
Let φ:(Y,NY)→(X,WX). As τ(φ(x))=τ(φ(z))=τ(φ(y))
contradicts x.y∈NY, either τ(φ(x))=τ(φ(z))
or τ(φ(y))=τ(φ(z)),
hence φ decomposes as φ=φ′∘ψz.y1
or φ=φ′∘ψz.x1.
∎
3.2 The Core functor
The flowing easy claim leads to some very helpful observations.
Claim 3.16**.**
Let Γ→Δ be a surjective graph
morphism. Let Γ′ be a graph resulting from folding two edges
in Γ, and Δ′ the pushout of Γ′←Γ→Δ
(which is obtained by folding the images of these edges in Δ,
if they are different from one another [MR695906]). Then Γ′→Δ′
is surjective.
Let Γ be a finite Y-labeled graph. One obtains CoreΓ
by a finite sequence of folding and trimming, and one can perform
foldings first and only then trimmings, as when a folded graph is
trimmed it remains folded. Following the claim, if Γ→Δ
is a surjective graph morphism and CoreΓ is obtained from
Γ without trimming then CoreΓ→CoreΔ is also
surjective.
Corollary 3.17**.**
If Γ is a core graph and (φ,Γ)
a stencil, then Fφ(Γ) is a core
graph, namely, CoreFφ(Γ)=Fφ(Γ).
If Γ,Δ are core graphs, (φ,Γ)
is a stencil and Γ→Δ is onto, then CoreFφ(Γ)→CoreFφ(Δ)
is onto.
We recall a Lemma from [MR3211804]:
Lemma 3.18** ([MR3211804]).**
Let Γ be a finite X-labeled graph such
that for every vertex v, except for possibly the base-point, there
are e,e′∈ι−1(v) with l(e)=l(e′). Then CoreΓ
is obtained from Γ by foldings alone (i.e. without trimming).
Corollary 3.19**.**
Let H≤K≤FY be subgroups such that there
is a non-trivial cyclically reduced word in H. If the morphism
ΓY(H)→ΓY(K) is onto then
ΓY(uHu−1)→ΓY(uKu−1)
is onto for every u∈FY.
Proof.
Let u∈FY be a reduced word. We construct the graph ΓYu(H)
by taking the graph ΓY(H)⊔Γu, gluing
len(u)+1∈V(Γu) to the base point of ΓY(H)
and setting the new base point to be 1∈V(Γu).
We notice that π1(ΓYu(H))=uHu−1
and that ΓYu(H)→ΓYu(K)
is onto. Since H and K contain a cyclically reduce word the
degrees of the base points of ΓY(H) and ΓY(K)
are at least two. From this we conclude that both ΓYu(H)
and ΓYu(K) satisfy the conditions of Lemma
3.18. Finally an inductive application of Claim 3.16
gives the result.
∎
4 Basis-independent surjectivity
Returning to Conjectures 1.1-1.3, recall
that for H≤K≤FY, we seek to show that for every free
extension FX of FY and every φ∈Aut(FX)
the graph morphism ΓX(φ(H))→ΓX(φ(K))
is surjective. Without loss of generality we may take X, from now
on, to be a fixed countably infinite set, and assume Y⊆X.
In Theorem 1.4 we even consider the set of non-degenerate
homomorphisms Hom((Y,∅),(X,WX)):
this includes all injective homomorphisms FY↪FX,
which in turn include all the restrictions of automorphisms of FX.
Using Remark 3.5 we translate the problem to showing
that all the graph morphisms {CoreFφ(ΓY(H)→ΓY(K))∣φ∈Hom((Y,∅),(X,WX))}
are surjective. This is a special case of the following problem.
Problem 4.1**.**
Given a morphism Γ→Δ
between two U-labeled core graphs, and an FGR
object (U,NU), we call the pair (Γ→Δ,(U,NU))
a surjectivity problem. We say that the problem
*resolves positively *if **all the morphisms
in the set P={CoreFφ(Γ→Δ)∣φ∈Hom((U,NU),(X,WX))}
are surjective. Note that taking (U,NU)=(Y,∅) we recover
the problem of surjectivity w.r.t. every non-degenerate φ:FU→FX,
and taking (U,NU)=(Y,W(Γ)) we obtain a trivial problem.
Let us give a simple criterion for showing that one instance of Problem
4.1 is contained in another, or that the
problems are equivalent. For U′-labeled core graphs Γ′,Δ′
and ψ:(U,NU)→(U′,NU′) an
FGR morphism such that Fψ(Γ→Δ)=Γ′→Δ′,
[TABLE]
so that the problem (Γ′→Δ′,(U′,NU′))
is contained in (Γ→Δ,(U,NU)).
If ψ is an FGR-isomorphism, then the two problems are equivalent.
Remark 4.2**.**
Following corollary 3.19
it is enough to consider surjectivity problems where π1(Γ)
contains a cyclically reduced word: otherwise, one can examine the
graph of a conjugate of π1(Γ) that contains
a cyclically reduced word.
Definition 4.3**.**
Let Γ be a Y-labeled folded graph. An FGR object (Y,NY)
is said to be a stencil space of Γ if W(Γ)⊆NY.
The reason for the name is that for any object (X,NX)
and morphism φ∈Hom((Y,NY),(X,NX)),
the pair (φ,Γ) is a stencil.
We show a method for determining whether all the morphisms in a problem
(Γ→Δ,(U,NU)) are surjective.
We distinguish three cases:
-
Γ→Δ is not surjective: clearly P resolves
negatively.
2. 2.
Γ→Δ is surjective and W(Γ)⊆NU,
i.e. (U,NU) is a stencil space of Γ: following
Corollary 3.17, P resolves positively.
3. 3.
Γ→Δ is surjective and W(Γ)\NU=∅:
in this case we cannot resolve P immediately. We call
this an ambiguous case.
When (Γ→Δ,(U,NU)) is an ambiguous
case, we cover it by five new problems, according to the partition
of Hom presented in §3.1: For any x.y∈W(Γ)\NU
we have
[TABLE]
where the last equality follows from Remark 3.5. Thus
we obtain five different subproblems \big{(}\text{Core}\mathcal{F}_{\psi_{z.y}^{i}}\left(\Gamma\to\Delta\right),(U_{i},N_{U_{i}})\big{)},
and P resolves positively iff all the five subproblems
do so. If one of these five problems is of case 1 then (Γ→Δ,(U,NU))
resolves negatively, if all are of case 2 then (Γ→Δ,(U,NU))
resolves positively, and otherwise we continue recursively and split
each ambiguous case into five sub-subproblems. We now give a toy example
to show how this process may end.
Example 4.4**.**
Let P be the surjectivity problem
with U={x,y}, NU={x.y−1,x−1.y,y−1.x−1,x.x−1},
Γ=ΓU(⟨xyx−1y−1⟩)
(the commutator) and Δ=ΓU(⟨x,y⟩).
The extension ⟨xyx−1y−1⟩<⟨x,y⟩
is algebraic so we know that this surjectivity problem resolves positively.
We will show how one can conclude this using the method presented
above. We notice that Γ↠Δ and x.y∈W(Γ)\NU,
so P is an ambiguous case. We split into five subproblems
(some of the calculations are explicit and some omitted and left to
the reader):
-
P1: ψx.y1=Id, U1=U,NU1={x.y−1,x−1.y,y−1.x−1,x.x−1,x.y},
Γ1=Γ, Δ1=Δ. The problem P1
resolves positively as (U1,NU1) is a stencil
space of Γ1 and Γ1↠Δ1.
2. 2.
P2: ψx.y2:x↦xty↦yt, U2={x,y,t}, NU2={t.y−1,x−1.t,y−1.x−1,x.t−1,y.t−1,x.y}
,Γ2=CoreFψx.y2(Γ)=ΓU2(ψx.y2⟨xyx−1y−1⟩)=ΓU2(⟨xtyx−1t−1y−1⟩),Δ2=ΓU2(⟨xt,yt⟩).
Again P2 resolves positively as (U2,NU2)
is a stencil space of Γ2 and Γ2↠Δ2.
3. 3.
P3: ψx.y3:x↦xyy↦yx, U3=U, NU3={x.y−1,x−1.x,y−1.x−1,y.x−1},
Γ3=Γ↠Δ=Δ3, and x.y∈W(Γ3)\NU3,
so this is an ambiguous case.
4. 4.
P4: ψx.y4:x↦xyy↦yx, U4=U, NU4={y.y−1,x−1.y,y−1.x−1,x.y−1},
Γ4=Γ↠Δ=Δ4, and x.y∈W(Γ4)\NU4
so this is an ambiguous case.
5. 5.
This case does not occur because x−1.y−1∈NU.
We notice first that the FGR-isomorphism x↔y gives
an equivalence between P4 and P3,
and second, that P3 is identical to the original P!
Now, it may seem that we are trapped in an infinite self-referential
loop: P resolves positively if P resolves
positively. Thankfully, Theorem 3.14 rescues us
from this nightmare. Let φ∈Hom((U,NU),(X,WX)).
We decompose φ by x.y recursively, mirroring the splitting
of the four subproblems. Because the decomposition of the morphism
via FGR is finite, it can loop through problems 3 and 4 only a
finite number of times until it arrives at a stencil space equivalent
to P1 or P2. Since P1
and P2 resolve positively, we conclude that so does
P.
Not all surjectivity problems end with equivalent ambiguous problems.
In some cases this process produces more and more different ambiguous
problems and therefore the algorithm does not end with a conclusive
answer. In the case of the counterexample (§5),
after a fair amount of splitting we end up with ambiguous cases equivalent
to or contained in problems we have previously encountered. In Section
§6 we discuss a property of the graph Γ
in a problem P that is a necessary condition for this
process to end.
4.1 Change of coordinates
With X and (Y,NY) as before, in P={CoreFφ(Γ→Δ)∣φ∈Hom((Y,NY),(X,WX))}
there can be many repetitions, as different homomorphisms φ
may give rise to the same graph morphism CoreFφ(Γ→Δ).
We can exploit this in a way which shortens calculations significantly
in §5.
Definition 4.5**.**
A change of coordinates consists of
FGR objects (V,NV),(U1,N1),…,(Un,Nn),
a group homomorphism σ:FV→FY (not necessarily
an FGR morphism),
an FGR morphism ψi:(V,NV)→(Ui,Ni) for every
i,
an FGR morphism σi:(Y,NY)→(Ui,Ni) for every
i,
[TABLE]
\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\sigma_{i}}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{i}}$$\scriptstyle{\sigma}$$\textstyle{X}$$\textstyle{U_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\prime}}
such that
-
π1(Δ)≤Imσ,
2. 2.
σ∗:φ↦φ∘σ takes Hom((Y,NY),(X,WX))
to Hom((V,NV),(X,WX)),
3. 3.
σi∘σ=ψi for every i,
4. 4.
for every φ∈Hom((Y,NY),(X,WX)) there
is an index i and a morphism φ′∈Hom((Ui,Ni),(X,WX))
such that φ∘σ=φ′∘ψi.
Proposition 4.6**.**
The problem (Γ→Δ,(Y,NY))
resolves positively if and only if (CoreFσi(Γ→Δ),(Ui,Ni))
resolves positively for every i.
Note that not all FGR morphisms in Hom((Y,NY),(X,WX))
necessarily factor through one of the σi, meaning there
could be φ∈Hom((Y,NY),(X,WX))
(as is the case in §5) with no decomposition
as φ=φ′∘σi – this is what makes the new
set of problems potentially simpler.
Proof.
Let π1(Γ)=H and π1(Δ)=K.
The morphisms ψi induce functions ψi∗:Hom((Ui,Ni),(X,WX))→Hom((V,NV),(X,WX)),
and conditions 2,3,4 imply 5:Imσ∗=⋃iImψi∗.
As Γ→Δ means that H≤K, for every homomorphism
φ:FY→FX we have φ(H)≤φ(K),
hence there is a unique graph morphism ΓX(φ(H))→ΓX(φ(K)).
We notice that:
[TABLE]
5 The counterexample
In this section we prove Theorem 1.4. showing that H=⟨bbaba−1⟩
and K=⟨b,aba−1⟩ constitute a counterexample to
Conjectures 1.1, 1.2, 1.3.
Theorem** (1.4).**
The extension H≤K is not algebraic, but for every morphism
φ:F{a,b}→FX with X arbitrary
and φ(a),φ(b)=1, the graph morphism ΓX(φ(H))→ΓX(φ(K))
is surjective.
First, H is a proper free factor of K, so in particular H≤K
is not an algebraic extension. Following Section §4,
we need to show that the problem (Γ{a,b}(H)→Γ{a,b}(K),({a,b},∅))
resolves positively. By a change of coordinate (Proposition 4.6)
we replace this problem with eight problems that are easier to analyze.
Let (V,NV)=({α,β},∅),
and
[TABLE]
We notice that H<K≤Imσ. For any non-degenerate φ:F{a,b}→FX,
the words φ(b)=φ∘σ(α)
and φ(aba−1)=φ∘σ(β)
are conjugate, hence there exist reduced words x,y,u,v∈FX
such that φ(b)=x⋅u⋅v⋅x−1,
φ(aba−1)=y⋅v⋅u⋅y−1
(in particular, vu and uv
are cyclically reduced). By non-degeneracy we can also assume u=1,
and if v=1 then u is cyclically reduced.
We perform a change of coordinates according to the eight possible
cases, with (Ui,Ni), ψi and σi being:
x
y
v
Ui
Ni
ψi(α),ψi(β)
σi(a),σi(b)
1
=1
=1
=1
{u}
{u.u−1}
u,u
u,u
2
=1
=1
=1
{y,u}
{y.u−1,u.y,u.u−1}
u,yuy−1
y,u
3
=1
=1
=1
{x,u}
{x.u−1,u.x,u.u−1}
xux−1,u
x−1,xux−1
4
=1
=1
=1
{x,u,y}
{y.u−1,u.y,y.xx.u−1,u.x,u.u−1,}
xux−1,yuy−1
yx−1,xux−1
5
=1
=1
=1
{v,u}
{v.u−1,u.v−1}
uv,vu
u−1,uv
6
=1
=1
=1
{v,u,y}
{v.u−1,u.v−1,y.v−1,u.y}
uv,yvuy−1
yv,uv
7
=1
=1
=1
{v,u,x}
{v.u−1,u.v−1,x.u−1,v.x}
xuvx−1,vu
u−1x−1,xuvx−1
8
=1
=1
=1
{v,u,y,x}
{v.x,y.v−1,u.yv.u−1,u.v−1,x.u−1,}
xuvx−1,yvuy−1
yu−1x−1,xuvx−1
For every 1≤i≤8 we denote Γi=Γ(σi(H))
and Δi=Γ(σi(K)). We
obtain eight problems, and we proceed to split them and identify all
the stencil spaces. We index the cases in the following manner: if
Case i is not a stencil space it splits into five cases Case i.1,i.2,…i.5.
For each case, the morphism ψi.j is the folding morphism
this subset of morphisms factors through. The co-domain of ψi.j
is the FGR object (Ui.j,Ni.j), and the graphs are indexed
by Γi.j=CoreFψi.j(Γi)
and Δi.j=CoreFψi.j(Δi),
and the Δ graphs are depicted in Figure 5.1.
In each case either W(Γi)\Ni is empty, hence
(Ui,Ni) is a stencil space of Γi and we can resolve
the subproblem, or it is not empty, and then we continue splitting.
Remark 4.2 allows us to conjugate, or
equivalently to change the base points of the graphs.
- Case 5.
Here Γ5=Γ(⟨uvuvvu⟩)↠Δ5
and W(Γ5)\N5={u.u−1,v.v−1}.
We split by u.u−1.
2. Case 5.1.
ψ5.1=ψu.u−11, N5.1={v.u−1,u.v−1,u.u−1},
Γ5.1=Γ5 and Δ5.1=Δ5 (hence Γ5.1↠Δ5.1).
W(Γ5.1)\NU5.1={v.v−1} (so
we must split by v.v−1).
3. Case 5.1.1.
ψ5.1.1=ψv.v−11, N5.1.1={v.u−1,u.v−1,u.u−1,v.v−1}.
Again Γ5=Γ5.1.1,Δ5.1.1=Δ5 and W(Γ5.1)\N5.1=∅,
hence the subcase resolves positively.
4. Case 5.1.2.
ψ5.1.2=ψv.v−12, v↦t−1vt,
N5.1.2={t.u−1,u.t,u.u−1,v.t−1,t−1.v−1,v.v−1};
Γ5.1.2=Γ(⟨ut−1vtut−1vvtu⟩)↠Δ5.1.2
and W(Γ5.1.2)\N5.1.2=∅
hence the subcase resolves positively.
5. Case 5.2.
ψ5.2=ψu.u−12, u↦t−1ut,
N5.2={v.t,u.t−1,u.u−1,t−1.u−1}; Γ5.2=Γ(⟨t−tutvt−1utvvt−1ut⟩)
is not cyclically reduced so we conjugate and set Γ5.2=Γ(⟨utvt−1utvvt−1u⟩)↠Δ5.2
and W(Γ5.2)\N5.2={v.v−1}.
We split by v.v−1.
6. Case 5.2.1.
ψ5.1=ψv.v−11, NU5.2.1={v.t,u.t−1,u.u−1,t−1.u−1,v.v−1};
Γ5.2.1=Γ5.2,Δ5.2.1=Γ5.2 and W(Γ5.2.1)\N5.2.1=∅,
hence the subcase resolves positively.
7. Case 5.2.2.
ψ5.2.2=ψv.v−12, v↦svs−1,
N5.2.2={s.t,u.t−1,u.u−1,t−1.u−1,v.s−1,s−1.v−1,v.v−1};
Γ5.2.2=Γ(⟨uts−1vst−1uts−1vvst−1u⟩)↠Δ5.2.2
and W(Γ5.2.2)\N5.2.2, hence the
subcase resolves positively.
8. Case 2.
Here Γ2=Γ(⟨uuyuy−1⟩)↠Δ2
and W(Γ2)\N2={u−1.y−1,u.y−1}.
We notice that u.u−1∈N2, so we can use the triangle rule
(Proposition 3.15) to deduce that every morphism factors
through either ψu−1.y−11 or ψu.y−11.
9. Case 2.1.
We have two cases: ψ2.1=ψu−1.y−11
and N2.1={y.u−1,u.y,u.u−1,u−1.y−1}, or ψ2.1′=ψu.y−11
and N2.1′={y.u−1,u.y,u.u−1,u.y−1}. These cases are
equivalent, as γ:(U2,N2.1)→(U2,N2.1′)
defined by u↦u−1,y↦y satisfies Fγ(Γ2.1→Δ2.1)=Γ2.1′→Δ2.1′.
Finally, Γ2.1=Γ2,Δ2.1=Δ2, and
W(Γ2.1)\N2.1={u.y−1}.
10. Case 2.1.1.
ψ2.1.1=ψu.y−11 and N2.1.1={y.u−1,u.y,u.u−1,u−1.y−1,u.y−1};
Γ2.1.1=Γ2.1,Δ2.1.1=Δ2.1 and W(Γ2.1.1)\N2.1.1=∅,
hence the subcase resolves positively.
11. Case 2.1.2.
ψ2.1.2=ψu.y−12, u↦ut,y↦t−1y
and N2.1.2={t.u−1,u−1.y,t.y,t.u−1,u.t−1,y−1.t−1,u.y−1};
Γ2.1.2=Γ(⟨utuyuty−1t⟩)↠Δ2.1.2
and W(Γ2.1.2)\N2.1.2=∅,
hence the subcase resolves positively.
12. Case 2.1.3.
ψ2.1.3=ψu.y−13, u↦uy−1,
N2.1.3={y.u−1,y−1.y,y−1.u−1,u.y} and Γ2.1.3=Γ(⟨uy−1uuy−1y−1⟩)↠Δ2.1.3.
We notice that γ:(U5,N5)→(U2.1.3,N2.1.3)
defined by u↦u−1,v↦y satisfies Fγ(Γ5→Δ5)
is conjugate to Γ2.1.3→Δ2.1.3, which implies
that this subproblem is contained in Case 5.
13. Case 2.1.4.
ψ2.1.4=ψu.y−14, y↦u−1y
and N2.1.4={y.u−1,u.y,u.u−1,u−1.y−1}; N2.1.4=N2.1
and Γ2.1.4=Γ2.1,Δ2.1.4=Δ2.1, so
this is the same problem as Case 2.1.
14. Case 3.
Here Γ3=Γ(⟨xuux−1u⟩)↠Δ3
and W(Γ3)\N3={x−1.u−1,u.x−1}
We notice that u.u−1∈N2 so we can split using the triangle
rule.
15. Case 3.1.
We have two cases ψ3.1=ψu−1.x−11,
N3.1={x.u−1,u.x,u.u−1,u.x−1} and ψ3.1′=ψu.x−11,
N3.1′={x.u−1,u.x,u.u−1,u−1.x−1}. These cases are
equivalent, as γ:(U3,N3.1)→(U3,N3.1′)
defined by u↦u−1,x↦x satisfies Fγ(Γ3.1→Δ3.1)=Γ3.1′→Δ3.1′.
Finally Γ3.1=Γ3,Δ3.1=Δ3 and W(Γ3.1)\N3.1={u−1.x−1}.
We split by u−1.x−1.
16. Case 3.1.1.
ψ3.1.1=ψu−1.x−11 and N3.1.1={x.u−1,u.x,u.u−1,u.x−1,x−1.u−1};
Γ3.1.1=Γ3.1,Δ3.1.1=Δ3.1 and W(Γ3.1.1)\N3.1.1=∅
hence the subcase resolves positively.
17. Case 3.1.2.
ψ3.1.2=ψu−1.x−12, x↦t−1x,u↦t−1u
and N3.1.2={x.t,u.x,u.t,t−1.x−1,t−1.u−1,x−1.u−1};
Γ3.1.2=Γ(⟨t−1xt−1ut−1ux−1u⟩)↠Δ3.1.2
and W(Γ3.1.2)\N3.1.2=∅,
hence the subcase resolves positively.
18. Case 3.1.3.
ψ3.1.3=ψu−1.x−13, x↦ux
and N3.1.3={x.u−1,u.x,u.u−1,u.x−1}; N3.1.3=N3.1
and Γ3.1.3=Γ3.1,Δ3.1.1=Δ3.1. This
is the same problem as Case 3.1.
19. Case 3.1.4.
ψ3.1.4=ψu−1.x−14, u↦xu
and N3.1.4={x.x−1,u.x,u.x−1,x.u−1}; Γ3.1.4=Γ(⟨xxuxuu⟩)↠Δ3.1.4.
We notice that γ:(U5,N5)→(U3.1.4,N3.1.4)
defined by u↦x,v↦u satisfies that Fγ(Γ5→Δ5)
is conjugate to Γ3.1.4→Δ3.1.4, hence this is
contained in Case 5.
20. Case 4.
Here Γ4=(⟨xuux−1yuy−1⟩)↠Δ4
and W(Γ4)\N4={x−1.y−1}.
We split by x−1.y−1∈W(Γ4)\N4.
21. Case 4.1.
ψ4.1=ψv.v−11, N4.1={x.u−1,u.x,u.u−1,y.u−1,u.y,y.x,x−1.y−1};
Γ4.1=Γ4,Δ4.1=Δ4 and W(Γ4.1)\N4.1=∅,
hence the subcase resolves positively.
22. Case 4.2.
ψ4.2=ψx−1.y−12, x↦t−1x,
y↦t−1y and N4.2={x.u−1,u.x,u.u−1,y.u−1y.u,y.x,x−1.t−1,y−1.t−1,y−1.x−1};
after conjugation Γ4.2=Γ4, Δ4.2=Δ4
and W(Γ4.2)\N4.2=∅, hence
the subcase resolves positively.
23. Case 4.3.
ψ4.3=ψx−1.y−13, x↦yx
and N4.3={x.u−1,u.x,u.u−1,y.u−1,u.y,y.x,y.x−1}.
We notice that γ:(U3,N3)→(U4.3,N4.3)
defined by u↦u,x↦x satisfies Fγ(Γ3→Δ3)=Γ4.3→Δ4.3,
hence this problem is contained in Case 3.
24. Case 4.4.
ψ4.4=ψx−1.y−14, y↦xy
and N4.4={x.u−1,u.x,u.u−1,y.u−1,u.y,y.x,x.y−1}.
We notice that γ:(U2,N2)→(U4.4,N4.4)
defined by u↦u,y↦y satisfies Fγ(Γ2→Δ2)=Γ4.4→Δ4.4,
hence this problem is contained in Case 2.
25. Case 6.
Here Γ6=(⟨uvuvyvuy−1⟩)↠Δ6
and W(Γ6)\N6={y−1.u−1,v.y−1}.
We notice that v.u−1∈N2 so we can split using the triangle
rule.
26. Case 6.1.
We have two cases ψ6.1=ψu−1.y−11,
N6.1={v.u−1,u.v−1,y.v−1,u.y,y−1.u−1} and ψ6.1′=ψv.y−11,
N6.1′={v.u−1,u.v−1,y.v−1,u.y,v.y−1}. These cases
are equivalent, as γ:(U6,N6.1)→(U6,N6.1′)
defined by u↦v−1,v↦u−1 satisfies Fγ(Γ6.1→Δ6.1)=Γ6.1′→Δ6.1′.
Γ6.1=Γ6,Δ6.1=Δ6 and W(Γ6.1)\N6.1={v.y−1}.
We split by v.y−1.
27. Case 6.1.1.
ψ6.1.1=ψv.y−11 and N6.1.1={v.u−1,u.v−1,y.v−1,u.y,y−1.u−1,v.y−1};
Γ6.1.1=Γ6,Δ6.1.1=Γ6 and W(Γ6.1.1)\N6.1.1=∅,
hence the subcase resolves positively.
28. Case 6.1.2.
ψ6.1.2=ψv.y−12, v↦vt,y↦t−1y
and N6.1.2={t.u−1,u.v−1,y.v−1,u.y,v.t−1,t−1.y−1};
Γ6.1.2=Γ(⟨uvtuvyvtuy−1t⟩)↠Δ6.1.2
and W(Γ6.1.2)\N6.1.2=∅,
hence the subcase resolves positively.
29. Case 6.1.3.
ψ6.1.3=ψv.y−13, v↦vy−1,
N6.1.3={y−1.u−1,u.v−1,y.v−1,u.y,y−1.u−1,v.y}
and Γ6.1.3=Γ(⟨y−1uvy−1uvvy−1u⟩).
We notice that γ:(U5,N5)→(U6.1.3,N6.1.3)
defined by u↦y−1u,v↦v satisfies Fγ(Γ5→Δ5)
is conjugate to Γ6.1.3→Δ6.1.3 therefore this
problem is contained in Case 5.
30. Case 6.1.4.
ψ6.1.4=ψv.y−14, y↦v−1y,
N6.1.4={v.u−1,u.v−1,y.v−1,u.y,v−1.y−1} and
Γ6.1.4(⟨vuvuyvuy−1⟩).
We notice that γ:(U2,N2)→(U6.1.4,N6.1.4)
defined by u↦vu satisfies Fγ(Γ2→Δ2)=Γ6.1.4→Δ6.1.4,
hence this problem is contained in Case 2.
31. Case 7.
Here Γ7=(⟨xuvuvx−1vu⟩)↠Δ7
and W(Γ7)\N7={x−1.v−1,u.x−1}.
We notice that u.v−1∈N2 so we can split using the triangle
rule.
32. Case 7.1
We have two cases ψ7.1=ψu.x−11, N7.1={v.u−1,u.v−1,x.u−1,v.x,x−1.u}
and ψ7.1′=ψx−1.v1, , N7.1′={v.u−1,u.v−1,x.u−1,v.x,x−1.v−1}.
These cases are equivalent, as γ:(U7,N7.1)→(U7,N7.1′)
defined by u↦v−1,v↦u−1 satisfies Fγ(Γ7.1→Δ7.1)=Γ7.1′→Δ7.1′;
Γ7.1=Γ7,Δ7.1=Γ7 and W(Γ7.1)\N7.1={x−1.v−1}.
We split by x−1.v−1.
33. Case 7.1.1.
ψ7.1.1=ψx−1.v−11 and N7.1.1={v.u−1,u.v−1,x.u−1,v.x,x−1.u,x−1.v−1};
Γ7.1.1=Γ7,Δ7.1.1=Δ7 and W(Γ7.1.1)\N7.1.1=∅,
hence the subcase resolves positively.
34. Case 7.1.2.
ψ7.1.2=ψx−1.v−12, x↦t−1x,v↦t−1v
and N7.1.2={v.u−1,u.t,x.u−1,v.x,t−1.x−1,t−1.v−1,x−1.v−1};
Γ7.1.2=Γ(⟨ut−1vut−1vx−1vut−1xu−1⟩)↠Δ7.1.2
and W(Γ7.1.2)\N7.1.2=∅,
hence the subcase resolves positively.
35. Case 7.1.3
ψ7.1.3=ψx−1.v−14, x↦vx,
N7.1.3={v.u−1,u.v−1,x.u−1,v.x,v.x−1} and Γ7.1.3(⟨xuvuvx−1uv⟩).
We notice that γ:(U3,N3)→(U7.1.3,N7.1.3)
defined by u↦uv,x↦x satisfies Fγ(Γ3→Δ3)=Γ7.1.3→Δ7.1.3,
hence this problem is contained in Case 3.
36. Case 7.1.4
ψ7.1.3=ψx−1.v−14, v↦xv,
N7.1.4={v.u−1,u.x−1,x.u−1,v.x,x.v−1} Γ7.1.4=(⟨uxvuxvvux⟩).
We notice that γ:(U5,N5)→(U7.1.4,N7.1.4)
defined by u↦ux,v↦v satisfies Fγ(Γ5→Δ5)=Γ7.1.4→Δ7.1.4,
hence this problem is contained in Case 5.
37. Case 8.
Here Γ8(xuvuvx−1yvuy−1)↠Δ8
andW(Γ8)\N8={x−1.y−1}.
We split by x−1.y−1.
38. Case 8.1.
ψ8.1=ψx−1.y−11, N8.1={v.u−1,u.v−1,x.u−1,v.x,y.v−1,u.y,x−1.y−1};
Γ8.1=Γ8,Δ8.1=Δ8 and W(Γ8.1)\N8.1=∅
hence the subcase resolves positively.
39. Case 8.2.
ψ8.2=ψx−1.y−12, x↦t−1x,y↦t−1y
and N8.2={v.u−1,u.v−1,x.u−1,v.x,y.v−1,u.y,t−1.y−1,t−1.x−1,x−1.y−1};
Γ8.2,Δ8.2 are conjugate to Γ8,Δ8
and W(Γ8.2)\N8.2=∅ hence
the subcase resolves positively.
40. Case 8.3.
ψ8.3=ψx−1.y−13, x↦yx
and N8.3={v.u−1,u.v−1,x.u−1,v.x,y.v−1,u.y,y.x−1}.
We notice that γ:(U7,N7)→(U8.3,N8.3)
defined by u↦u,x↦x,v↦v satisfies Fγ(Γ7→Δ7)=Γ8.3→Δ8.3,
hence this problem is contained in Case 7.
41. Case 8.4.
ψ8.4=ψx−1.y−14, y↦xy
and N8.4={v.u−1,u.v−1,x.u−1,v.x,y.v−1,u.y,x.y−1}.
We notice that γ:(U6,N6)→(U8.4,N8.4)
defined by u↦u,y↦y,v↦v satisfies Fγ(Γ6→Δ6)=Γ8.4→Δ8.4,
hence this problem is contained in Case 6.
42. Case 8.5.
ψ8.5=ψx−1.y−15, y↦x
and N8.4={v.u−1,u.v−1,x.u−1,v.x,x.v−1,u.x}.
We notice that γ:(U5,N5)→(U8.4,N8.4)
defined by u↦u,v↦v satisfies Fγ(Γ5→Δ5)=Γ8.5→Δ8.5,
hence this problem is contained in Case 5.
43. Case 1.
We notice that Γ1↠Δ1
and that W(Γ1)\N1=∅,
hence the case resolves positively.
6 Stencil-finiteness
Let P={Γ→Δ∣(U,NU)} be
a surjectivity problem. In this section we define and examine a property
of the graph Γ, that is necessary for the recursive splitting
of P (as described in §4) to
end in a finite set of equivalent ambiguous cases. Examining Example
4.4 we see that all the graphs {CoreFφ(Γ(xyx−1y−1))∣φ∈Hom((U,NU),(X,WX))}
are contained in two stencil spaces {Fφ(Γ(xyx−1y−1))∣φ∈Hom((U1,NU1),(X,WX))}
and {Fφ(Γ(xtyx−1t−1y−1))∣φ∈Hom((U2,NU2),(X,WX))}.
This is a reformulation of a classic result of Wicks [MR0142610]
about the commutator, as we explain below. A closer examination of
the computation in §5 shows that there
are five stencil spaces (up to conjugation) that include all other
stencil spaces: Cases 1, 2.1, 5.1.1, 5.1.2, and 6.1. The set of graphs
{CoreFφΓ(⟨bbaba−1⟩)∣φ∈Hom(({a,b},∅),(X,WX))}
is the union of the stencil spaces ⋃i∈I{Fφ(Γi)∣φ∈Hom((Ui,NUi),(X,WX))}
with I={1,2.1,5.1.1,5.1.2,6.1} the set of indices of
stencil space cases. This fact, formulated differently, appears in
[MR285589] (where Meskin attributes it to unpublished work by
Newman). In fact, he shows more generally that this is true for all
Baumslag-Solitar relators blabma−1 (l,m∈Z).
Technically this generalizes Wicks’ result, as the commutator has
l=1,m=−1. This leads us to define the notion of stencil-finiteness
below. Even though Meskin and Wicks did not define this notion, each
of them showed that some word satisfies it. They were interested in
words w with this property because it implies an easy algorithm
to determine whether there is a solution to the equation w(y1,…,yn)=a
with a∈FX and unknowns y1,…,yn∈FY. First we define
the following poset:
Definition 6.1**.**
Let X be a countably infinite set, and define an equivalence relation
between folded X-labeled graphs: Γ∼Δ iff there
exists non-degenerate homomorphisms φ,ψ:FX→FX
such that (φ,Γ) and (ψ,Δ)
are stencils and Fφ(Γ)=Δ,
Fψ(Δ)=Γ. On the set of equivalence
classes we define [Γ]≤[Δ] iff there exists a non-degenerate
homomorphism φ:FX→FX such that (φ,Δ)
is a stencil and Fφ(Δ)=Γ.
Definition 6.2**.**
We say that a folded Y-labeled graph Δ
has stencil-finiteness if the set {[Core(Fφ(Δ))]∣φ∈Hom((Y,∅),(X,WX))}
has a finite number of maximal elements.
This property can be translated to a property of words, subgroups
and homomorphisms of free groups by taking the corresponding graphs.
Not all words have this property: Using the methods described in this
paper one can show that the word x3y2 does not have this
property. This is a lengthy computation and not the subject matter
of the paper, and is therefore omitted. Is stencil finiteness a sufficient
condition for the process ending? We do not know the answer. A priori,
the graph Γ may have stencil finiteness while Δ doesn’t,
and while the graphs CoreFφ(Γ)
could repeat themselves, the graphs CoreFφ(Δ)
can be distinct and thus lead to infinite distinct ambiguous cases.
We do not have an explicit example for this mainly because there aren’t
many graphs known to have stencil finiteness. The words known to have
stencil finiteness are blabma−1 with l,m∈Z
(this includes the commutator), primitive words. One can show that
powers of a word with this property have it as well.
We end with some open questions:
-
Are there any other words with this property? Are there core graphs
of subgroups with rank ≥2 that have this property?
2. 2.
Is a surjectivity problem P={Γ→Δ∣(U,NU)}
such that Γ has stencil finiteness decidable?
3. 3.
Is stencil finiteness a decidable property?
4. 4.
Is there a purely algebraic interpretation of the property of an extension
being onto on every base?
References