Geometry & Topology

Indicability, residual finiteness, and simple subquotients of groups acting on trees

Pierre-Emmanuel Caprace and Phillip Wesolek

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We establish three independent results on groups acting on trees. The first implies that a compactly generated locally compact group which acts continuously on a locally finite tree with nilpotent local action and no global fixed point is virtually indicable; that is to say, it has a finite-index subgroup which surjects onto . The second ensures that irreducible cocompact lattices in a product of nondiscrete locally compact groups such that one of the factors acts vertex-transitively on a tree with a nilpotent local action cannot be residually finite. This is derived from a general result, of independent interest, on irreducible lattices in product groups. The third implies that every nondiscrete Burger–Mozes universal group of automorphisms of a tree with an arbitrary prescribed local action admits a compactly generated closed subgroup with a nondiscrete simple quotient. As applications, we answer a question of D Wise by proving the nonresidual finiteness of a certain lattice in a product of two regular trees, and we obtain a negative answer to a question of C Reid, concerning the structure theory of locally compact groups.

Article information

Source
Geom. Topol., Volume 22, Number 7 (2018), 4163-4204.

Dates
Received: 22 August 2017
Accepted: 13 April 2018
First available in Project Euclid: 14 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.gt/1544756697

Digital Object Identifier
doi:10.2140/gt.2018.22.4163

Mathematical Reviews number (MathSciNet)
MR3890774

Zentralblatt MATH identifier
06997386

Subjects
Primary: 20E08: Groups acting on trees [See also 20F65]
Secondary: 22D05: General properties and structure of locally compact groups

Keywords
trees lattices in products locally compact groups

Citation

Caprace, Pierre-Emmanuel; Wesolek, Phillip. Indicability, residual finiteness, and simple subquotients of groups acting on trees. Geom. Topol. 22 (2018), no. 7, 4163--4204. doi:10.2140/gt.2018.22.4163. https://projecteuclid.org/euclid.gt/1544756697


Export citation

References

  • R Alperin, Locally compact groups acting on trees, Pacific J. Math. 100 (1982) 23–32
  • U Bader, B Duchesne, J Lécureux, Amenable invariant random subgroups, Israel J. Math. 213 (2016) 399–422
  • I Bondarenko, B Kivva, Automaton groups and complete square complexes, preprint (2017)
  • M Burger, S Mozes, Groups acting on trees: from local to global structure, Inst. Hautes Études Sci. Publ. Math. 92 (2000) 113–150
  • M Burger, S Mozes, Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. 92 (2000) 151–194
  • P-E Caprace, T De Medts, Simple locally compact groups acting on trees and their germs of automorphisms, Transform. Groups 16 (2011) 375–411
  • P-E Caprace, N Monod, Isometry groups of non-positively curved spaces: discrete subgroups, J. Topol. 2 (2009) 701–746
  • P-E Caprace, N Monod, Decomposing locally compact groups into simple pieces, Math. Proc. Cambridge Philos. Soc. 150 (2011) 97–128
  • P-E Caprace, N Monod, A lattice in more than two Kac–Moody groups is arithmetic, Israel J. Math. 190 (2012) 413–444
  • H Furstenberg, Boundary theory and stochastic processes on homogeneous spaces, from “Harmonic analysis on homogeneous spaces”, Amer. Math. Soc., Providence, RI (1973) 193–229
  • A Garrido, Y Glasner, S Tornier, Automorphism groups of trees: generalities and prescribed local actions, from “New directions in locally compact groups” (P-E Caprace, N Monod, editors), London Math. Soc. Lecture Note Ser. 447, Cambridge Univ. Press (2018) 92–116
  • E Hewitt, K A Ross, Abstract harmonic analysis, I: Structure of topological groups, integration theory, group representations, 2nd edition, Grundl. Math. Wissen. 115, Springer (1979)
  • J Huang, Commensurability of groups quasi-isometric to RAAGs, Invent. Math. 213 (2018) 1179–1247
  • D Janzen, D T Wise, A smallest irreducible lattice in the product of trees, Algebr. Geom. Topol. 9 (2009) 2191–2201
  • J S Kimberley, G Robertson, Groups acting on products of trees, tiling systems and analytic $K``$–theory, New York J. Math. 8 (2002) 111–131
  • A Le Boudec, Groups acting on trees with almost prescribed local action, Comment. Math. Helv. 91 (2016) 253–293
  • C Nebbia, Amenability and Kunze–Stein property for groups acting on a tree, Pacific J. Math. 135 (1988) 371–380
  • N Radu, New simple lattices in products of trees and their projections, preprint (2017)
  • M S Raghunathan, Discrete subgroups of Lie groups, Ergeb. Math. Grenzgeb. 68, Springer (1972)
  • D A Rattaggi, Computations in groups acting on a product of trees: normal subgroup structures and quaternion lattices, PhD thesis, Eidgenössische Technische Hochschule Zürich (2004) Available at \setbox0\makeatletter\@url www.rattaggi.ch/phd.pdf \unhbox0
  • D Rattaggi, Anti-tori in square complex groups, Geom. Dedicata 114 (2005) 189–207
  • C D Reid, Dynamics of flat actions on totally disconnected, locally compact groups, New York J. Math. 22 (2016) 115–190
  • J-P Serre, Trees, Springer (1980)
  • J Tits, Sur le groupe des automorphismes d'un arbre, from “Essays on topology and related topics” (A Haefliger, R Narasimhan, editors), Springer (1970) 188–211
  • P Wesolek, Elementary totally disconnected locally compact groups, Proc. Lond. Math. Soc. 110 (2015) 1387–1434
  • D T Wise, Non-positively curved squared complexes: aperiodic tilings and non-residually finite groups, PhD thesis, Princeton University (1996) Available at \setbox0\makeatletter\@url https://search.proquest.com/docview/304259249 {\unhbox0
  • D T Wise, Complete square complexes, Comment. Math. Helv. 82 (2007) 683–724