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

Pierre-Emmanuel Caprace and Tom De Medts (UGent)
(2011) TRANSFORMATION GROUPS. 16(2). p.375-411
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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.
Keywords
Primitive group, Tree, Spheromorphisms, Germ of automorphism, Branch group, Commensurator, ROOTED TREES, Totally disconnected locally compact group, Simple group

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Citation

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MLA
Caprace, Pierre-Emmanuel, and Tom De Medts. “Simple Locally Compact Groups Acting on Trees and Their Germs of Automorphisms.” TRANSFORMATION GROUPS, vol. 16, no. 2, 2011, pp. 375–411, doi:10.1007/s00031-011-9131-z.
APA
Caprace, P.-E., & De Medts, T. (2011). Simple locally compact groups acting on trees and their germs of automorphisms. TRANSFORMATION GROUPS, 16(2), 375–411. https://doi.org/10.1007/s00031-011-9131-z
Chicago author-date
Caprace, Pierre-Emmanuel, and Tom De Medts. 2011. “Simple Locally Compact Groups Acting on Trees and Their Germs of Automorphisms.” TRANSFORMATION GROUPS 16 (2): 375–411. https://doi.org/10.1007/s00031-011-9131-z.
Chicago author-date (all authors)
Caprace, Pierre-Emmanuel, and Tom De Medts. 2011. “Simple Locally Compact Groups Acting on Trees and Their Germs of Automorphisms.” TRANSFORMATION GROUPS 16 (2): 375–411. doi:10.1007/s00031-011-9131-z.
Vancouver
1.
Caprace P-E, De Medts T. Simple locally compact groups acting on trees and their germs of automorphisms. TRANSFORMATION GROUPS. 2011;16(2):375–411.
IEEE
[1]
P.-E. Caprace and T. De Medts, “Simple locally compact groups acting on trees and their germs of automorphisms,” TRANSFORMATION GROUPS, vol. 16, no. 2, pp. 375–411, 2011.
@article{1989519,
  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.}},
  author       = {{Caprace, Pierre-Emmanuel and De Medts, Tom}},
  issn         = {{1083-4362}},
  journal      = {{TRANSFORMATION GROUPS}},
  keywords     = {{Primitive group,Tree,Spheromorphisms,Germ of automorphism,Branch group,Commensurator,ROOTED TREES,Totally disconnected locally compact group,Simple group}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{375--411}},
  title        = {{Simple locally compact groups acting on trees and their germs of automorphisms}},
  url          = {{http://doi.org/10.1007/s00031-011-9131-z}},
  volume       = {{16}},
  year         = {{2011}},
}
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