Groups Acting on Trees With Prescribed Local Action

Abstract

We extend Burger-Mozes theory of closed, non-discrete, locally quasiprimitive automorphism groups of locally finite, connected graphs to the semiprimitive case, and develop a generalization of Burger-Mozes universal groups acting on the regular tree T d of degree d. Three applications are given: First, we characterize the Banks-Elder-Willis k-closures of locally transitive subgroups of Aut(T d ) containing an involutive inversion, and thereby partially answer two questions raised by Banks-Elder-Willis. Second, we offer a new perspective on the Weiss conjecture. Third, we obtain a characterization of the automorphism types which the quasi-center of a non-discrete subgroup of Aut(T d ) may feature in terms of the group's local actions. In doing so, we explicitly construct closed, non-discrete, compactly generated subgroups of Aut(T d ) with non-trivial quasi-center, thereby answering a question of Burger, and show that Burger-Mozes theory does not generalize to the transitive case.

Unitary representations of totally disconnected locally compact groups satisfying Ol'shanskii's factorization

“…Amoung other things, Radu completely classified them and proved that they are k-closed for some constant k ∈ N depending on the group which implies that they satisfy the independence property IP k . Other families of groups on which Theorem C apply could be constructed using the generalisation of Buger-Mozes groups described by Tornier in [Tor20]. However, for those groups some considerations have to be made on the local action described on the k-balls in order to satisfy the hypothesis H q for some given q ∈ N. Finally, the existence criteria for irreducible representation with depth l for the generic filtrations S q will be discussed in Section 4.3.…”

Homomorphic images of locally compact groups acting on trees and buildings

“…Next, using the notation of universal groups as set out in [20], we prove the following two results. Since the generalised universal groups U k (F) satisfy Property P k [20,Proposition 3.7] and do not stabilise any proper non-empty subtree or fix an end of T , this also gives us the following.…”

“…Next, using the notation of universal groups as set out in [20], we prove the following two results. Since the generalised universal groups U k (F) satisfy Property P k [20,Proposition 3.7] and do not stabilise any proper non-empty subtree or fix an end of T , this also gives us the following. The theorem also allows us to show that the universal groups U 1 (F) satisfy the closed range property when F is transitive and generated by point stabilisers.…”

“…The hypotheses of Corollary 4.8 imply that U 1 (F) + 1 is non-trivial. On the other hand, the group U k (F) in Corollary 4.7 may be discrete, see[20,Proposition 3.12], and so it may happen that U k (F) + k is trivial.…”

Homomorphic Images of Locally Compact Groups Acting on Trees and Buildings

“…Next, using the notation of universal groups as set out in [Tor20], we prove the following two results. Since the generalised universal groups U k pF q satisfy Property P k [Tor20, Proposition 3.7] and do not stabilise any proper non-empty subtree or fix an end of T , this also gives us the following.…”