Simple locally compact groups acting on trees and their germs of automorphisms

Abstract

Automorphism groups of locally finite trees provide a large class of examples of simple totally disconnected locally compact groups. It is desirable to understand the connections between the global and local structure of such a group. Topologically, the local structure is given by the commensurability class of a vertex stabiliser; on the other hand, the action on the tree suggests that the local structure should correspond to the local action of a stabiliser of a vertex on its neighbours. We study the interplay between these different aspects for the special class of groups satisfying TitsE1/4 independence property. We show that such a group has few open subgroups if and only if it acts locally primitively. Moreover, we show that it always admits many germs of automorphisms which do not extend to automorphisms, from which we deduce a negative answer to a question by George Willis. Finally, under suitable assumptions, we compute the full group of germs of automorphisms; in some specific cases, these turn out to be simple and compactly generated, thereby providing a new infinite family of examples which generalise NeretinE1/4s group of spheromorphisms. Our methods describe more generally the abstract commensurator group for a large family of self-replicating profinite branch groups

Compact presentability of tree almost automorphism groups

“…Here we give an outline of their construction (see Section 4 for precise definitions). Every finite permutation subgroup These groups appear in [13], where a careful study of the abstract commensurator group of self-replicating profinite wreath branch groups is carried out (we refer to [2] for an introduction to abstract commensurators of profinite groups). Let W k (D) be the closed subgroup of Aut(T d,k ) fixing pointwise the first level of T d,k and acting by an element of W (D) in each subtree rooted at level one.…”

Dense normal subgroups and chief factors in locally compact groups

“…Let be the Burger–Mozes universal simple group for the alternating group on 6 elements with the usual permutation representation (see for a discussion of these groups). The group acts on the 6‐regular tree .…”

“…Just as with compact generation, discreteness is not a normal compression invariant, even if we restrict to compressions between non-abelian characteristically simple groups; cf. i∈Z Alt (5) and i∈Z Alt (5). There is a more general property that is close to discreteness and admits better preservation properties: An l.c.s.c.…”

Computing the Scale of an Endomorphism of a totally Disconnected Locally Compact Group

“…Neretin groups are also studied in [28], where it is shown that they do not contain a lattice, and the notation used here conforms with that paper. The papers [29], on abstract commensurators, and [30], on 'germs of automorphisms' are also relevant. Neretin's groups are also the inspiration for the simple groups acting on trees recently constructed in [31].…”

“…Since abstract t.d.l.c. groups always contain a compact open, and hence commensurated, subgroup, they may be realised concretely as groups of commensurators quite generally, see [29,30,35]. Moreover, given any pair (G, H) such that H is a commensurated subgroup of G, a t.d.l.c.…”