This article is an expanded version of the talks given by the authors at the Arbeitsgemeinschaft "Totally Disconnected Groups", held at Oberwolfach in October 2014. We recall the basic theory of automorphisms of trees and Tits' simplicity theorem, and present two constructions of tree groups via local actions with their basic properties: the universal group associated to a finite permutation group by M. Burger and S. Mozes, and the k-closures of a given group by C. Banks, M. Elder and G. Willis.

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.

The concepts of the scale and tidy subgroups for an automorphism of a totally disconnected locally compact group were defined in seminal work by George A. Willis in the 1990s, and recently generalized to the case of endomorphisms (G. A. Willis, Math. Ann. 361 (2015), 403-442). We show that central facts concerning the scale, tidy subgroups, quotients, and contraction groups of automorphisms extend to the case of endomorphisms. In particular, we obtain results concerning the domain of attraction around an invariant closed subgroup. (f) The compact open subgroups W ⊆ lev(α) with α(W ) = W form a basis of identity neigbourhoods in lev(α).If G is a Lie group over a totally disconnected local field (as in [3] and [18]) and α : G → G is an analytic endomorphism with small tidy subgroups, then con(α), lev(α) and con − (α) are Lie subgroups of G (in the strong sense of submanifolds) and the product map in (2) is an analytic diffeomorphism, see [9].By (e) in Theorem F, the automorphism α| lev(α) is distal, see [14]. Information concerning contractive automorphisms of locally compact groups can be found in [19] and [10]; contractive analytic automorphisms of Lie groups over a totally disconnected local field K are discussed in [21] (for K = Q p ) and [8].

This article concerns Burger-Mozes universal groups acting on regular trees locally like a given permutation group of finite degree. We also consider locally isomorphic generalizations of the former due to Le Boudec and Lederle. For a large class of such permutation groups and primes p we determine their local p-Sylow subgroups as well as subgroups of their p-localization, which is identified as a group of the same type in certain cases.Date: March 23, 2020.