Characterization of non-compact locally compact groups by cocompact subgroups

Bilel Kadri

doi:10.1515/jgth-2019-0029

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Characterization of non-compact locally compact groups by cocompact subgroups

Published/Copyright: August 20, 2019

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From the journal Journal of Group Theory Volume 23 Issue 1

Abstract

A subgroup H of a topological group G is called cocompact (or uniform) if the quotient space G/H¯ is compact, where H¯ denotes the closure of H in G. The purpose of this paper is to give a characterization of non-compact locally compact groups with the property that every non-trivial closed (respectively, open) subgroup is cocompact (respectively, has finite index).

1 Introduction and main results

1.1 Motivation

In the investigation of the structure of locally compact groups, the occurrence of closed subgroups is unavoidable. Note that one of the most important problems is to characterize a group in terms of closed subgroups. This problem has attracted considerable attention over many years, and many works have appeared in this direction. In the present paper, our primary motivation comes from the papers of D. L. Armacost and S. A. Morris [1, 5, 6, 7, 9]. In fact, they provided various characterizations of familiar groups.

1.2 Groups with compact quotients

We start by introducing the basic notion in the paper.

Definition 1.1 (Cocompact subgroup).

A subgroup H of a topological group G is called cocompact (or uniform) if the quotient space G/H¯ is compact, where H¯ denotes the closure of H in G.

We say that a locally compact group is just-non-compact, or briefly a JNC group, if it is non-compact and every non-trivial closed normal subgroup is cocompact [10]. Groups of this sort were studied by P.-E. Caprace and N. Monod. In order to present their results, we need to recall that a topological group is monolithic with monolith L if the intersection of all non-trivial closed normal subgroups is itself a non-trivial group L. In [2], they gave a description of JNC groups as follows.

Theorem 1.2 ([2, Theorem E]).

Let G be a compactly generated JNC group. Then one of the following holds.

G is monolithic, and its monolith is a quasi-product with finitely many isomorphic topologically simple groups as quasi-factors.
G is monolithic with monolith L isomorphic to ℝn. Moreover, G/L is isomorphic to a closed irreducible subgroup of O⁢(n). In particular, G is an almost connected Lie group.
G is discrete and residually finite.

Proposition 1.3 ([2, Proposition 2.4]).

Let G be a compactly generated JNC group. If G admits a non-trivial discrete normal subgroup, then G is either discrete or Rn-by-finite.

In this paper, motivated by JNC groups, we will restrict our attention to locally compact groups with the following property:

(CQ) every non-trivial closed subgroup is cocompact.

A topological group satisfying the property (CQ) will often be called CQ-group. In view of Theorem 1.2, it makes sense to consider the following problem.

Problem 1.4.

Characterize non-compact locally compact CQ-groups.

The following is the first main result of the paper.

Theorem A (Theorem 3.9 below).

A non-compact locally compact CQ-group is isomorphic either to Z or to R.

1.3 Groups with finite quotients

Let ℤp denote the topological group of p-adic integers for a prime number p. In [8], S. A. Morris, S. Oates-Williams and H. B. Thompson gave a characterization of ℤp in terms of its closed subgroups as follows.

Theorem 1.5 ([8, Theorem 1]).

Let G be a non-discrete locally compact group. Then the following are equivalent.

Every non-trivial closed subgroup has finite index.
G is isomorphic to ℤp.

Since closed subgroups of finite index are open, the result mentioned previously leaves open the following general problem.

Problem 1.6.

Characterize non-compact locally compact groups with the property that every non-trivial open subgroup has finite index.

These facts, in analogy with CQ-groups, permit us to introduce the following definition.

Definition 1.7 (FQ-group).

A non-compact locally compact group G is called an FQ-group if every non-trivial open subgroup has finite index in G.

Now, we complete the picture by providing the second main result in this paper.

Theorem B (Theorem 4.6 below).

Let G be a non-compact locally compact group. Then the following are equivalent.

G is an FQ-group.
G is either isomorphic to ℤ or almost connected.

2 Some remarkable groups

In this section, we summarize some preliminary results about locally compact groups which will be needed later. For more details and proofs, we refer the reader to the book of M. Stroppel [12].

Definition 2.1 (Almost connected group).

A topological group G is called almost connected if the factor group G/G0 of G modulo the connected component G0 of the identity is compact.

Theorem 2.2.

Every almost connected locally compact group G is homeomorphic to Rn×C, where C is a compact subgroup of G.

Proof.

See [3, Theorem 8.6]. ∎

Definition 2.3 (Totally disconnected topological space).

A topological space is said to be totally disconnected if every connected component consists of a single point.

Proposition 2.4.

Let G0 be the identity component of a topological group G. Then the following conclusions hold.

G is totally disconnected if and only if G0 is trivial.
The quotient group G/G0 is totally disconnected.

The following result is fundamental in the theory of totally disconnected locally compact groups.

Theorem 2.5 (Van Dantzig’s theorem).

Let G be a totally disconnected locally compact group. Then the set of compact open subgroups of G is a basis of identity neighborhoods.

Proof.

See [13] or [12, Proposition 4.13] . ∎

3 Characterization of CQ-groups

Proposition 3.1.

If G is any CQ-group (not necessarily locally compact), then it is either almost connected or totally disconnected.

Proof.

If G0 is the connected component of the identity, then G0 is either the trivial group or G/G0 is compact. In the second case, G is almost connected. ∎

Definition 3.2 (Compact-free group).

A topological group is called compact-free if it has no compact subgroup except the trivial one.

The following result plays a crucial role in the sequel. Despite its simplicity, it is enormously useful.

Proposition 3.3.

If G is a non-compact CQ-group, then it is compact-free.

Proof.

Assume that G contains a non-trivial compact subgroup K. Then both K and G/K are compact, which implies that G is also compact. This contradicts our assumption on G. ∎

Proposition 3.4.

If G is a non-compact CQ-group which is locally compact and totally disconnected, then it is discrete and every non-trivial subgroup of G has finite index.

Proof.

According to Van Dantzig’s theorem (Theorem 2.5), the locally compact totally disconnected group G has a basis of compact open subgroups. But, in view of Proposition 3.3, G is compact-free, and consequently it is discrete. On the other hand, if H is any non-trivial subgroup of G, then the quotient space G/H is compact and discrete and thus finite. ∎

The following theorem was proved by Y. Fedorov in 1951, appears in [11, p. 446] and appears also with an elementary proof in [4] by C. Lanski.

Theorem 3.5.

An infinite group G is isomorphic to the group of integers Z if and only if every non-trivial subgroup of G has finite index.

Proposition 3.6.

Every non-compact locally compact totally disconnected CQ-group is isomorphic to Z.

Proof.

This follows immediately from Proposition 3.4 and Theorem 3.5. ∎

Remark 3.7.

Let G be a non-compact CQ-group and n∈ℕ. Then the following two assertions are equivalent.

Gn is a CQ-group.
n=1.

Proposition 3.8.

If G is an almost connected non-compact non-discrete locally compact CQ-group, then it is isomorphic to R.

Proof.

According to Theorem 2.2, the group G is homeomorphic to ℝn×C, where C is a compact subgroup of G. But G is compact-free, so C is the trivial group, and G is a homeomorphic to ℝn. As G is a CQ-group, we obtain that n=1, and therefore G is isomorphic to ℝ. ∎

Finally, combining Propositions 3.6 and 3.8, we obtain the following.

Theorem 3.9.

A non-compact locally compact CQ-group is isomorphic either to Z or to R.

4 Characterization of FQ-groups

As a first step towards the second main result, we restrict ourselves to the case of total disconnectedness.

4.1 Totally disconnected FQ-groups

Proposition 4.1.

The class of FQ-groups is closed under passing to quotients modulo proper closed normal subgroups.

Proof.

Let N be a proper closed normal subgroup of an FQ-group G, and let π:G→G/N denote the canonical projection; pick H a non-trivial open subgroup of G/N. Since G is an FQ-group, the open subgroup π-1⁢(H) has finite index in G. On the other hand, we have the isomorphism (G/π-1⁢(H))≅(G/N)/H. Then H has finite index in G/N. Hence the quotient G/N is also an FQ-group. ∎

Remark 4.2.

An FQ-group cannot have proper compact open subgroups.

Proposition 4.3.

Let G be an FQ-group. Then the following assertions are equivalent.

G is totally disconnected.
G is discrete.
G is isomorphic to ℤ.

Proof.

(1) ⇒ (2): On one hand, according to Van Dantzig’s theorem (see Theorem 2.5), the set of compact open subgroups of G is a basis of identity neighborhoods. On the other hand, G does not have proper compact open subgroups. Thus the trivial subgroup must be open, and therefore G is discrete. (2) ⇒ (3): For discrete groups, it is obvious that the notions FQ and CQ are equivalent. Then, by applying Proposition 3.6, we obtain that G is isomorphic to ℤ. The implication (3) ⇒ (1) is evident. ∎

4.2 Non-totally disconnected FQ-groups

Lemma 4.4.

Let G be a non-compact locally compact group. If G is almost connected, then it is an FQ-group.

Proof.

Let H be a proper open subgroup of G. It is clear that G0⊂H. Then H is cocompact since G is almost connected. On the other hand, we have that G/H is discrete. Hence H has finite index in G, and therefore G is an FQ-group. ∎

Proposition 4.5.

Let G be an FQ-group. Then the following assertions are equivalent.

G is not totally disconnected.
G is almost connected.

Proof.

For the first implication, we have that if G is not totally disconnected, then its identity component G0 is not trivial. Then the quotient G/G0 is a totally disconnected FQ-group. Aiming for a contradiction, assume that G/G0 is not compact. According to Proposition 4.3, we have that G/G0 is isomorphic to ℤ, and therefore G0 is an open subgroup of G. On the other hand, we recall that G is an FQ-group. Then G0 has finite index in contradiction to the obvious fact that ℤ is not compact. The second implication follows immediately from the non-compactness of G. ∎

Combining Propositions 4.3 and 4.5, we obtain the next result.

Theorem 4.6.

Let G be a non-compact locally compact group. Then the following are equivalent.

G is an FQ-group.
G is either isomorphic to ℤ or almost connected.

In conclusion, it seems natural to consider the following question:

Question 4.7.

Characterize non-compact locally compact groups with the property that every proper discrete subgroup is cocompact.

Communicated by George Willis

Funding statement: This work was completed with the support of General Direction of the Scientific Research and Technological Renovation (DGRSRT) Research Laboratory (Tunisia) Grant number LR 11ES52.

Acknowledgements

It is a great pleasure to thank Professor Hatem Hamrouni for his valuable remarks that help to increase the quality of the paper. The author is also grateful to the referee for his constructive comments and suggestions which greatly helped to improve the proof of the first main theorem.

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Received: 2019-02-12

Revised: 2019-07-22

Published Online: 2019-08-20

Published in Print: 2020-01-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2019-0029