On the Component Factor Group G/G_0 of a Pro-Lie Group G

TL;DR

This paper proves that a pro-Lie group is almost connected if all its finite-dimensional Lie group quotients have finitely many components, filling a gap in the understanding of pro-Lie group structure.

Contribution

It establishes a new characterization of almost connected pro-Lie groups based on their Lie group quotients, which was previously unproven.

Findings

Pro-Lie groups are almost connected if all Lie group quotients have finitely many components.

The proof involves verifying the completeness of the totally disconnected factor group.

The result clarifies the relationship between pro-Lie groups and their finite-dimensional quotients.

Abstract

A pro-Lie group $G$ is a topological group such that $G$ is isomorphic to the projective limit of all quotient groups $G/N$ (modulo closed normal subgroups $N$) such that $G/N$ is a finite dimensional real Lie group. A topological group is almost connected if the totally disconnected factor group $G_t:= G/G_0$ of $G$ modulo the identity component $G_0$ is compact. In this case it is straightforward that each Lie group quotient $G/N$ of $G$ has finitely many components. However, in spite of a comprehensive literature on pro-Lie groups, the following theorem, proved here, was not available until now: A pro-Lie group $G$ is almost connected if each of its Lie group quotients $G/N$ has finitely many connected components. The difficulty of the proof is the verification of the completeness of $G_t$.

Full text

Projective Limits of Almost Connected Lie Groups

A notorious problem in the structure theory of pro-Lie groups is the completeness of quotient groups, notably that of the group $G/G_{0}$ of connected components. In one of the sources on pro-Lie groups, [2], the section following Definition 4.24 on pp.195ff. exhibits some of the characteristic difficulties involving the completeness of quotients of a pro-Lie group $G$ in general and the quotient $G_{t}=G/G_{0}$ in particular. In their entirety, these difficulties remain unresolved today. We shall settle the completeness issue of $G_{t}$ here for any pro-Lie group whose Lie group quotients have finitely many components.

Existing literature (see [4], Corollary 8.4) provides the following conclusion, which reinforces the independent interest in the result of this note:

An almost connected pro-Lie group $G$ contains a maximal compact subgroup $C$ and a closed subspace $V$ homeomorphic to $\mathbb{R}^{J}$ for a set $J$ such that

$(c,x)\mapsto cx\colon C\times V\to G$

is a homeomorphism.

So, let $G$ denote a topological group and ${\cal N}(G)$ the set of all closed normal subgroups of $G$ for which $G/N$ is a Lie group. With these conventions we formulate a theorem, to be proved subsequently. The proof requires some effort. It is based on information from [2].

{Theorem}

For a pro-Lie group $G$, the following statements are equivalent:

1. (1)

$G_{t}$ is compact, 2. (2)

There is a compact totally disconnected subspace $D\subseteq G$ being mapped homeomorphically onto $G_{t}$ by the quotient map $q_{t}\colon G\to G_{t}$. 3. (3)

$(G/N)_{t}$ is finite for all $N\in{\cal N}(G)$.

The proof of the theorem will require the proof of some new lemmas and some references to existing literature. The first one is cited from [4], Main Theorem 8.1, Corollary 8.3.

{Lemma}

Let $G$ be an almost connected pro-Lie group. Then the following conclusions hold:

1. (i)

$G$ contains a maximal compact subgroup $C$, and any compact subgroup of $G$ has a conjugate inside $C$. 2. (ii)

$G=G_{0}C$. 3. (iii)

$G$ contains a profinite subgroup $D$ such that $G=G_{0}D$.

For every compact group $K$ there is a compact totally disconnected subspace $D\subseteq K$ such that $(k,d)\mapsto kd:K_{0}\times D\to K$ is a homeomorphism (see [3], Corollary 10.38, p. 573). From Lemma Projective Limits of Almost Connected Lie Groups we know that $G$ is almost connected iff there is a compact subgroup $K\subseteq G$ such that $G=G_{0}K$. Write $K=K_{0}D$ with the topological direct factor $D\subseteq K$ as we just pointed out. Then $G=G_{0}K_{0}D=G_{0}D$ and so $(g,d)\mapsto gd:G_{0}\times D\to G$ is readily seen to be a homeomorphism. Thus, in Theorem Projective Limits of Almost Connected Lie Groups, Condition (1) implies (2), and for (2)${\Rightarrow}$(1) there is nothing to prove.

Let us establish that (1)${\Rightarrow}$(3):

Assume $G/N$ to be a Lie group quotient of $G$. Then $(G/N)_{0}=G_{0}N/N\cong G_{0}/(G_{0}\cap N)$ (cf. [2], Lemma 3.29, p.152). Let $K$ be a compact subgroup of $G$ such that $G=G_{0}K$, and let $L=G/N$. Then $L=(G_{0}N/N)(KN/N)=L_{0}C$ for the compact Lie group $C=KN/N$. Thus $L_{t}=L_{0}C/L_{0}\cong C/(C\cap L_{0})$ is a compact totally disconnected Lie group and is therefore finite. This proves (3).

There remains a proof of the implication (3)${\Rightarrow}$(1):

For the moment let us assume that the following hypothesis is satisfied

(H) $G_{t}$ is a complete topological group.

By [2], Corollary 3.31, hypothesis (H) implies that $G_{t}$ is prodiscrete, that is, $G_{t}=\lim_{N\in{\cal N}(G_{t})}G_{t}/N$ where $G_{t}/N$ is discrete. Now $G_{t}/N$ is a Lie group quotient of $G$ and so is finite by (3). Hence $G_{t}$ is profinite and thus compact. This proves Condition (1).

It therefore remains to prove (H). For this purpose we shall invoke results from [2], pp.195ff.

Firstly, we define ${\cal M}(G)$ to be the subset of all $M\in{\cal N}(G)$ with the additional property that each open subgroup $N\subseteq M$ from ${\cal N}(G)$ has finite index in $M$. We shall then use

{Lemma}

If $G$ is a pro-Lie group such that

(*) each Lie group quotient $G/N$, ${N}\in{\cal N}(G)$ is almost connected,

then ${\cal M}(G)$ is cofinal in ${\cal N}({G})$ and thus is a filter basis.

Moreover, $G$ is the strict projective limit of the $G/M$, $M\in{\cal M}(G)$.

{Proof}

For the proof see Lemma 4.25 in [2], pp.195 and 196.

We note that Lemma 4.25 in [2] states as hypothesis that $G$ is almost connected which implies (*). But the hypothesis (*) is all that is used in the proof of Lemma 4.25.

Any set $\cal Z$ of subsets of a set $G$ may be considered as a subbasis of closed sets for a topology. If $G$ is a topological group, and $\cal Z$ is the set of all cosets $gM$ with $g\in G$ and $M\in{\cal M}(G)$, then $\cal Z$ generates the set of closed sets of a topology on $G$, called the Z-topology.

{Lemma}

The Z-topology on a pro-Lie group $G$ satisfying Condition (*) of Lemma Projective Limits of Almost Connected Lie Groups is a compact $T_{1}$-topology.

{Proof}

See Proposition 4.27 of [2], pp. 197–201.

Again we note that Proposition 4.27 in [2] assumes the hypothesis that $G$ is almost connected, but the proof of the conclusion of Lemma Projective Limits of Almost Connected Lie Groups only uses Hypothesis (*) of Lemma Projective Limits of Almost Connected Lie Groups.

We now adjust the proof of Theorem 4.28 on p. 202 of [2] for our purposes.

{Lemma}

Let $G$ be a pro-Lie group satisfying hypothesis (*) of Lemma Projective Limits of Almost Connected Lie Groups. Then $G_{t}$ is complete.

We note right away that Lemma Projective Limits of Almost Connected Lie Groups will prove hypothesis (H) and therefore complete the proof of Theorem Projective Limits of Almost Connected Lie Groups.

Proof of Lemma Projective Limits of Almost Connected Lie Groups. We let $f\colon G\to G_{t}=G/G_{0}$ be the quotient morphism and consider a Cauchy filter $\cal C$ on $G_{t}$. We have to show that $\cal C$ converges. By Lemma Projective Limits of Almost Connected Lie Groups, ${\cal M}(G)$ is cofinal in ${\cal N}(G)$. For each $N\in{\cal M}(G)$ let $N^{*}=\overline{f(N)}$ and let $p_{N^{*}}\colon G\to G_{t}{/N^{*}}$ be the quotient morphism. Then the image $p_{N^{*}}({\cal C})$ is a Cauchy filter in the Lie group $G_{t}/N^{*}$ and thus has a limit $g_{N}$. Then $(g_{N})_{N\in{\cal M}(G)}\in\prod_{N\in{\cal M}(G)}G_{t}/N^{*}$ is an element of $\lim_{N\in{\cal M}(G)}G_{t}/N^{*}$; indeed $\cal C$ has to converge to a point in the completion of $G_{t}$. Now let $F_{N}=(p_{N^{*}}\circ f)^{-1}({g}_{N})$. Then $\{F_{N}:N\in{\cal M}(G)\}$ is a filter basis consisting of cosets modulo $N_{*}\stackrel{{\scriptstyle\rm def}}{{=}}\overline{G_{0}N}$ of $G$. We claim that $N_{*}\in{\cal M}(G)$. Indeed we have $N_{*}\in{\cal N}(G)$. Now we let $M$ be an open subgroup of $N_{*}\supseteq G_{0}$. Then $G_{0}\subseteq M$, so $G_{0}N\subseteq MN$ and $MN$ is open-closed in $N_{*}=\overline{G_{0}N}$. Thus $N_{*}=MN$. So $MN/M\cong N/(N\cap M)$ is discrete and then $M\cap N$ is open in $N$. But $N\in{\cal M}(G)$ then implies that $N_{*}/M\cong N/(M\cap N)$ is finite. This shows that $N_{*}\in{\cal M}(G)$ as claimed. Since $G$ is Z-compact by Lemma Projective Limits of Almost Connected Lie Groups, we find an element $g\in\bigcap_{N\in{\cal M}(G)}F_{N}$. But then $p_{N^{*}}(f(g))={g}_{N}$ for all $N\in{\cal M}(G)$ which implies that $f(g)=\lim{\cal C}$. Thus every Cauchy filter in $G_{t}$ converges showing that $G_{t}$ is complete. ∎

This completes the proof of Theorem Projective Limits of Almost Connected Lie Groups.

An inspection of [2] shows that the following questions appear to be unsettled:

Question 1. For which pro-Lie groups $G$ is ${\cal M}(G)$ cofinal in ${\cal N}(G)$?

For each of these groups $G$ we would know that $G$ is (isomorphic to) the strict projective limit $\lim_{M\in{\cal M}(G)}G/M$. In [2] this is proved of all almost connected pro-Lie groups.

Test examples are the nondiscrete pro-discrete groups $\mathbb{Z}^{(\mathbb{N})}$ and $\mathbb{Z}^{\mathbb{N}}$ (see e.g. [2], Example 4.4ff., Proposition 5.2).

Question 2. For which pro-Lie groups is the Z-topology compact?

In [2], Proposition 4.27, this is shown for almost connected pro-Lie groups, and here we have proved it for those pro-Lie groups $G$ all of whose Lie group quotients are known to be almost connected.

Theorem Projective Limits of Almost Connected Lie Groups suggests the following rather general question:

Question 3. When is the projective limit of a projective system of almost connected topological groups almost connected?

Theorem Projective Limits of Almost Connected Lie Groups says that within the category of pro-Lie groups we have an affirmative answer for the projective system of all Lie group quotients. See also some background information in [2] in and around Theorem 1.27, p. 88.

One remark is in order in the context of the Z-topology discussed in [2] on pp. 197–203:

In Exercise E4.2(i), p. 202, it is pointed out that {\cal M}(\mathbb{Z})=\big{\{}\{0\},\mathbb{Z}\big{\}} and that therefore the Z-topology on $\mathbb{Z}$, being the cofinite topology, is compact. Whereas the topology generated by the set of cosets $z+N$, $N\in{\cal N}(\mathbb{Z})$, the set of all subgroups of $\mathbb{Z}$, fails to be compact.

Theorem Projective Limits of Almost Connected Lie Groups will play a significant role in the authors’ study [1] of weakly complete real or complex topological algebras with identity, which will explore in detail their relation to pro-Lie theory and aims for a systematic treatment of weakly complete group algebras of topological groups and their representation and duality theories.

Acknowledgment. The authors thank the referee for his swift, yet thorough contributions to the final form of this note.