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Simple locally compact groups acting on trees and their germs of automorphisms
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 Tits' 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 Neretin's group of spheromorphisms. Our methods describe more
generally the abstract commensurator group for a large family of
self-replicating profinite branch groups
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