caa a22 4500 445862599 CHVBK 20180317145448.0 cr unu---uuuuu 170323e20110601xx s 000 0 eng 10.1007/s00031-011-9131-z doi (NATIONALLICENCE)springer-10.1007/s00031-011-9131-z Simple locally compact groups acting on trees and their germs of automorphisms [Elektronische Daten] [Pierre-Emmanuel Caprace*, Tom De Medts] 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 Titsʼ 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 Neretinʼs group of spheromorphisms. Our methods describe more generally the abstract commensurator group for a large family of self-replicating profinite branch groups. Springer Science+Business Media, LLC, 2011 Tree nationallicence Totally disconnected locally compact group nationallicence Simple group nationallicence Primitive group nationallicence Spheromorphisms nationallicence Germ of automorphism nationallicence Branch group nationallicence Commensurator nationallicence Caprace* Pierre-Emmanuel UCLouvain, Chemin du Cyclotron 2, 1348, Louvain-la-Neuve, Belgium aut De Medts Tom Ghent University Department of Mathematics, Krijgslaan, 281-S22 9000, Gent, Belgium aut Transformation Groups SP Birkhäuser Verlag Boston 16/2(2011-06-01), 375-411 1083-4362 16:2<375 2011 16 31 https://doi.org/10.1007/s00031-011-9131-z text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00031-011-9131-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Caprace* Pierre-Emmanuel UCLouvain, Chemin du Cyclotron 2, 1348, Louvain-la-Neuve, Belgium aut NATIONALLICENCE 700 1- De Medts Tom Ghent University Department of Mathematics, Krijgslaan, 281-S22 9000, Gent, Belgium aut NATIONALLICENCE 773 0- Transformation Groups SP Birkhäuser Verlag Boston 16/2(2011-06-01), 375-411 1083-4362 16:2<375 2011 16 31 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer