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Besides automorphism groups of trees and buildings, I was wondering if the lattices in general totally disconnected locally compact groups have been studied in the literature? I appreciate if you introduce some references.

If you are acquainted with the structure theory of totally disconnected groups (due to George Willis), does this theory says anything about lattices in TDLC groups?

More generally, I am interested to know if there is a strategy to generalize certain results (as the above inquiry) from the automorphism groups of trees to the more general totally disconnected groups.

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Any discrete group is locally compact and is a lattice in itself. –  Misha Apr 4 at 17:59
    
@Misha: Is this as much as I can get? –  Vahid Shirbisheh Apr 4 at 18:13
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Yes, unless you get a more specific question. –  Misha Apr 4 at 19:43
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Given a finitely generated group $\Gamma$ and a Cayley graph $X$ for $\Gamma$, then $\Gamma$ is a uniform lattice in the automorphism group of $X$, which is a (not necessarily discrete) t.d.l.c. group. –  Colin Reid Apr 5 at 8:48
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As far as I know, the question of whether a given t.d.l.c. group has lattices is very difficult in general. Willis theory tells you a lot about individual automorphisms, but not so much about the global structure. –  Colin Reid Apr 5 at 9:01

2 Answers 2

If you assume that a TDG acts geometrically on a CAT(0) space, then something interesting can be said about lattices, see here and references therein to get started.

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It's quite a specific family of examples, but you might find this paper of interest as an example of how a compactly generated simple t.d.l.c. group can fail to have any lattices:

Bader, Caprace, Gelander and Mozes, 'Simple groups without lattices', Bull. Lond. Math. Soc. 44 Nr. 1, (2012), pp. 55–67. http://arxiv.org/abs/1008.2911

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The point in this paper is that the TDLC group is simple. Otherwise it's rather trivial to find TDLC groups without lattices, e.g. non-unimodular LC groups, or any LC group having $\mathbf{Q}_p^n$ as open subgroup. –  YCor Apr 5 at 15:05
    
I am wondering can one reduce simplicity with the condition that there is no open normal subgroup in the TDLC group? Is it still an interesting (and non-trivial) question to look for TDLC groups without lattices? –  Vahid Shirbisheh Apr 5 at 17:11
    
@Vahid: the direct product of two non-discrete topologically simple LC groups is not simple, although it contains no open normal subgroup except itself. –  YCor Apr 5 at 22:46
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@VahidShirbisheh, no, it won't be open (unless the other factor is finite). –  HJRW Apr 6 at 6:43
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@HJRW: Yes, you are right, unless the other summand is a discrete group. –  Vahid Shirbisheh Apr 6 at 6:54

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