A closed subgroup $H$ of a locally compact group $G$ is confined if the closure of the conjugacy class of $H$ in the Chabauty space of $G$ does not contain the trivial subgroup. We establish a dynamical criterion on the action of a totally disconnect
ed locally compact group $G$ on a compact space $X$ ensuring that no relatively amenable subgroup of $G$ can be confined. This property is equivalent to the fact that the action of $G$ on its Furstenberg boundary is free. Our criterion applies to the Neretin groups. We deduce that each Neretin group has two inequivalent irreducible unitary representations that are weakly equivalent. This implies that the Neretin groups are not of type I, thereby answering a question of Y.~Neretin.
We construct several series of explicit presentations of infinite hyperbolic groups enjoying Kazhdans property (T). Some of them are significantly shorter than the previously known shortest examples. Moreover, we show that some of those hyperbolic Ka
zhdan groups possess finite simple quotient groups of arbitrarily large rank; they constitute the first known specimens combining those properties. All the hyperbolic groups we consider are non-positively curved k-fold generalized triangle groups, i.e. groups that possess a simplicial action on a CAT(0) triangle complex, which is sharply transitive on the set of triangles, and such that edge-stabilizers are cyclic of order k.
Let $Gamma$ be a finitely generated group and $X$ be a minimal compact $Gamma$-space. We assume that the $Gamma$-action is micro-supported, i.e. for every non-empty open subset $U subseteq X$, there is an element of $Gamma$ acting non-trivially on $U
$ and trivially on the complement $X setminus U$. We show that, under suitable assumptions, the existence of certain commensurated subgroups in $Gamma$ yields strong restrictions on the dynamics of the $Gamma$-action: the space $X$ has compressible open subsets, and it is an almost $Gamma$-boundary. Those properties yield in turn restrictions on the structure of $Gamma$: $Gamma$ is neither amenable nor residually finite. Among the applications, we show that the (alternating subgroup of the) topological full group associated to a minimal and expansive Cantor action of a finitely generated amenable group has no commensurated subgroups other than the trivial ones. Similarly, every commensurated subgroup of a finitely generated branch group is commensurate to a normal subgroup; the latter assertion relies on an appendix by Dominik Francoeur, and generalizes a result of Phillip Wesolek on finitely generated just-infinite branch groups. Other applications concern discrete groups acting on the circle, and the centralizer lattice of non-discrete totally disconnected locally compact (tdlc) groups. Our results rely, in an essential way, on recent results on the structure of tdlc groups, on the dynamics of their micro-supported actions, and on the notion of uniformly recurrent subgroups.
We amend the statement of point~(i) in Theorem~1.3 in arxiv:0901.1022 and supply the additional arguments and minor changes for the results that depend on it. We also seize the occasion and generalize to non-finitely generated lattices.
In this note, we provide a complete description of the closed sets of real roots in a Kac-Moody root system.
Let $T_1, T_2$ be regular trees of degrees $d_1, d_2 geq 3$. Let also $Gamma leq mathrm{Aut}(T_1) times mathrm{Aut}(T_2)$ be a group acting freely and transitively on $VT_1 times VT_2$. For $i=1$ and $2$, assume that the local action of $Gamma$ on $T
_i$ is $2$-transitive; if moreover $d_i geq 7$, assume that the local action contains $mathrm{Alt}(d_i)$. We show that $Gamma$ is irreducible, unless $(d_1, d_2)$ belongs to an explicit small set of exceptional values. This yields an irreducibility criterion for $Gamma$ that can be checked purely in terms of its local action on a ball of radius~$1$ in $T_1$ and $T_2$. Under the same hypotheses, we show moreover that if $Gamma$ is irreducible, then it is hereditarily just-infinite, provided the local action on $T_i$ is not the affine group $mathbf F_5 rtimes mathbf F_5^*$. The proof of irreducibility relies, in several ways, on the Classification of the Finite Simple Groups.
Let $G$ be a group. A subset $F subset G$ is called irreducibly faithful if there exists an irreducible unitary representation $pi$ of $G$ such that $pi(x) eq mathrm{id}$ for all $x in F smallsetminus {e}$. Otherwise $F$ is called irreducibly unfait
hful. Given a positive integer $n$, we say that $G$ has Property $P(n)$ if every subset of size $n$ is irreducibly faithful. Every group has $P(1)$, by a classical result of Gelfand and Raikov. Walter proved that every group has $P(2)$. It is easy to see that some groups do not have $P(3)$. We provide a complete description of the irreducibly unfaithful subsets of size $n$ in a countable group $G$ (finite or infinite) with Property $P(n-1)$: it turns out that such a subset is contained in a finite elementary abelian normal subgroup of $G$ of a particular kind. We deduce a characterization of Property $P(n)$ purely in terms of the group structure. It follows that, if a countable group $G$ has $P(n-1)$ and does not have $P(n)$, then $n$ is the cardinality of a projective space over a finite field. A group $G$ has Property $Q(n)$ if, for every subset $F subset G$ of size at most $n$, there exists an irreducible unitary representation $pi$ of $G$ such that $pi(x) e pi(y)$ for any distinct $x, y$ in $F$. Every group has $Q(2)$. For countable groups, it is shown that Property $Q(3)$ is equivalent to $P(3)$, Property $Q(4)$ to $P(6)$, and Property $Q(5)$ to $P(9)$. For $m, n ge 4$, the relation between Properties $P(m)$ and $Q(n)$ is closely related to a well-documented open problem in additive combinatorics.
We study lattices in a product $G = G_1 times dots times G_n$ of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that $G_i$ is non-compact and every
closed normal subgroup of $G_i$ is discrete or cocompact (e.g. $G_i$ is topologically simple). We show that the set of discrete subgroups of $G$ containing a fixed cocompact lattice $Gamma$ with dense projections is finite. The same result holds if $Gamma$ is non-uniform, provided $G$ has Kazhdans property (T). We show that for any compact subset $K subset G$, the collection of discrete subgroups $Gamma leq G$ with $G = Gamma K$ and dense projections is uniformly discrete, hence of covolume bounded away from $0$. When the ambient group $G$ is compactly presented, we show in addition that the collection of those lattices falls into finitely many $Aut(G)$-orbits. As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor. We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-non-compact tdlc group $G$ is a Chabauty limit of discrete subgroups, then some compact open subgroup of $G$ is an infinitely generated pro-$p$ group for some prime $p$. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups.
This text is the preprint version of the concluding chapter for the book New Directions in Locally Compact Groups published by Cambridge University Press in the series Lecture Notes of the LMS. The recent progress on locally compact groups surveyed i
n that volume also reveals the considerable extent of the unexplored territories. Therefore, we wish to conclude it by mentioning a few open problems related to the material covered in the book and that we consider important at the time of this writing.
These notes are devoted to lattices in products of trees and related topics. They provide an introduction to the construction, by M. Burger and S. Mozes, of examples of such lattices that are simple as abstract groups. Two features of that constructi
on are emphasized: the relevance of non-discrete locally compact groups, and the two-step strategy in the proof of simplicity, addressing separately, and with completely different methods, the existence of finite and infinite quotients. A brief history of the quest for finitely generated and finitely presented infinite simple groups is also sketched. A comparison with Margulis proof of Knesers simplicity conjecture is discussed, and the relevance of the Classification of the Finite Simple Groups is pointed out. A final chapter is devoted to finite and infinite quotients of hyperbolic groups and their relation to the asymptotic properties of the finite simple groups. Numerous open problems are discussed along the way.
