Abstract
We study contraction groups for automorphisms of totally disconnected locally compact groups using the scale of the automorphism as a tool. The contraction group is shown to be unbounded when the inverse automorphism has non-trivial scale and this scale is shown to be the inverse value of the modular function on the closure of the contraction group at the automorphism. The closure of the contraction group is represented as acting on a homogenous tree and closed contraction groups are characterised.
Similar content being viewed by others
References
[Bou68] N. Bourbaki,Groupes et Algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968; publiées dans Actualités scientifiques et industriels 1337.
[BT65] A. Borel and J. Tits,Groupes réductifs, Publications mathématiques de l’Institut des Hautes Études Scientifiques27 (1965), 55–150.
[DD89] W. Dicks and M. J. Dunwoody,Groups acting on Graphs, Volume 17 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1989.
[DS91] S. G. Dani and R. Shah,Contraction subgroups and semistable measures on p-adic Lie groups, Mathematical Proceedings of the Cambridge Philosophical Society110 (1991), 299–306.
[Glö98a] H. Glöckner,Scale functions on linear groups over local skew fields, Journal of Algebra205 (1998), 525–541.
[Glö98b] H. Glöckner,Scale functions on p-adic Lie groups, Manuscripta Mathematica97 (1998), 205–215.
[GW02] H. Glöckner and G. A. Willis,Topologization of Hecke pairs and Hecke C *-algebras, Topology Proceedings26 (2001–2002), 565–591.
[Haz88] M. Hazewinkel (ed.),Encyclopaedia of Mathematics. Vol. 1. A-B, D. Reidel Publ. Co., Dordrecht, 1988. Translated from the Russian.
[HR79] E. Hewitt and K. A. Ross,Abstract Harmonic Analysis; Volume I, Volume 115 of Grundlehren der mathematischen Wissenschaften, second edition, Springer-Verlag, Berlin, 1979.
[HS88] W. Hazod and E. Siebert,Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups, Journal of Theoretical Probability1 (1988), 211–225.
[LR68] H. Leptin and L. Robertson,Every locally compact MAP group is unimodular, Proceedings of the American Mathematical Society19 (1968), 1079–1082.
[Mar89] G. A. Margulis,Discrete Subgroups of Semisimple Lie Groups, Volume 17 (3. Folge) of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, New York, 1989.
[Möl02] R. G. Möller,Structure theory of totally disconnected locally compact groups via graphs and permutations, Canadian Journal of Mathematics54 (2002), 795–827.
[Moo80] C. C. Moore,The Mautner phenomenon for general unitary representations, Pacific Journal of Mathematics86 (1980), 155–169.
[MR76] P. R. Müller-Römer,Kontrahierende Erweiterungen und kontrahierbare Gruppen, Journal für die reine und angewandte Mathematik283/284 (1976), 238–264.
[Pra82] G. Prasad,Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits, Bulletin de la Société Mathématique de France110 (1982), 197–202.
[RD81] W. Roelcke and S. Dierolf,Uniform Structures on Topological Groups and their Quotients, McGraw-Hill International Book Company, New York, 1981.
[Ros79] J. Rosenblatt,A distal property of groups and the growth of connected locally compact groups, Mathematika26 (1979), 94–98.
[Ser80] J.-P. Serre,Trees, Springer-Verlag, Berlin, 1980. The original French editionArbres, amalgames, SL 2 was published as Astérisque46 (by Société Mathématique de France) in 1977.
[Sie86] E. Siebert,Contractive automorphisms on locally compact groups, Mathematische Zeitschrift191 (1986), 73–90.
[Sie89] E. Siebert,Semistable convolution semigroups and the topology of contraction groups, inProbability Measures on Groups IX (Proceedings of a Conference held at Oberwolfach/FRG 1988), Lecture Notes in Mathematics1379, Springer-Verlag, Berlin, 1989, pp. 325–343.
[Wan84] J. S. P. Wang,The Mautner phenomenon for p-adic Lie groups, Mathematische Zeitschrift185 (1984), 403–412.
[Wil] G. A. Willis,Tidy subgroups for commuting automorphisms of totally disconnected locally compact groups: An analogue of simultaneous triangularisation of matrices, New York Journal of Mathematics10 (2004), 1–35. Available at <http://nyjm.albany.edu:8000/j/2004/Vol10.htm.<
[Wil94] G. A. Willis,The structure of totally disconnected locally compact groups, Mathematische Annalen300 (1994), 341–363.
[Wil01] G. A. Willis,Further properties of the scale function on a totally disconnected locally compact group, Journal of Algebra237 (2001), 142–164.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by DFG Grant BA 1966/1–1 and ARC Grant A69700321.
Rights and permissions
About this article
Cite this article
Baumgartner, U., Willis, G.A. Contraction groups and scales of automorphisms of totally disconnected locally compact groups. Isr. J. Math. 142, 221–248 (2004). https://doi.org/10.1007/BF02771534
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02771534