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

Caprace, Pierre-Emmanuel¹ ; Wesolek, Phillip²

¹ Institut de Recherche en Mathématiques et Physique, Université catholique de Louvain, Louvain-la-Neuve, Belgium
² Department of Mathematical Sciences, Binghamton University, Binghamton, NY, United States

Geometry & topology, Tome 22 (2018) no. 7, pp. 4163-4204.

Résumé

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 $\mathbb{Z}$. 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.

Détail

DOI : 10.2140/gt.2018.22.4163

Classification : 20E08, 22D05
Keywords: trees, lattices in products, locally compact groups

Bibliographie

[1] R Alperin, Locally compact groups acting on trees, Pacific J. Math. 100 (1982) 23 | DOI

[2] U Bader, B Duchesne, J Lécureux, Amenable invariant random subgroups, Israel J. Math. 213 (2016) 399 | DOI

[3] I Bondarenko, B Kivva, Automaton groups and complete square complexes, preprint (2017)

[4] M Burger, S Mozes, Groups acting on trees: from local to global structure, Inst. Hautes Études Sci. Publ. Math. 92 (2000) 113 | DOI

[5] M Burger, S Mozes, Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. 92 (2000) 151 | DOI

[6] P E Caprace, T De Medts, Simple locally compact groups acting on trees and their germs of automorphisms, Transform. Groups 16 (2011) 375 | DOI

[7] P E Caprace, N Monod, Isometry groups of non-positively curved spaces: discrete subgroups, J. Topol. 2 (2009) 701 | DOI

[8] P E Caprace, N Monod, Decomposing locally compact groups into simple pieces, Math. Proc. Cambridge Philos. Soc. 150 (2011) 97 | DOI

[9] P E Caprace, N Monod, A lattice in more than two Kac–Moody groups is arithmetic, Israel J. Math. 190 (2012) 413 | DOI

[10] H Furstenberg, Boundary theory and stochastic processes on homogeneous spaces, from: "Harmonic analysis on homogeneous spaces", Amer. Math. Soc. (1973) 193

[11] A Garrido, Y Glasner, S Tornier, Automorphism groups of trees: generalities and prescribed local actions, from: "New directions in locally compact groups" (editors P E Caprace, N Monod), London Math. Soc. Lecture Note Ser. 447, Cambridge Univ. Press (2018) 92 | DOI

[12] E Hewitt, K A Ross, Abstract harmonic analysis, I : Structure of topological groups, integration theory, group representations, 115, Springer (1979) | DOI

[13] J Huang, Commensurability of groups quasi-isometric to RAAGs, Invent. Math. 213 (2018) 1179 | DOI

[14] D Janzen, D T Wise, A smallest irreducible lattice in the product of trees, Algebr. Geom. Topol. 9 (2009) 2191 | DOI

[15] J S Kimberley, G Robertson, Groups acting on products of trees, tiling systems and analytic K“–theory, New York J. Math. 8 (2002) 111

[16] A Le Boudec, Groups acting on trees with almost prescribed local action, Comment. Math. Helv. 91 (2016) 253 | DOI

[17] C Nebbia, Amenability and Kunze–Stein property for groups acting on a tree, Pacific J. Math. 135 (1988) 371 | DOI

[18] N Radu, New simple lattices in products of trees and their projections, preprint (2017)

[19] M S Raghunathan, Discrete subgroups of Lie groups, 68, Springer (1972)

[20] 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)

[21] D Rattaggi, Anti-tori in square complex groups, Geom. Dedicata 114 (2005) 189 | DOI

[22] C D Reid, Dynamics of flat actions on totally disconnected, locally compact groups, New York J. Math. 22 (2016) 115

[23] J P Serre, Trees, Springer (1980) | DOI

[24] J Tits, Sur le groupe des automorphismes d’un arbre, from: "Essays on topology and related topics" (editors A Haefliger, R Narasimhan), Springer (1970) 188

[25] P Wesolek, Elementary totally disconnected locally compact groups, Proc. Lond. Math. Soc. 110 (2015) 1387 | DOI

[26] D T Wise, Non-positively curved squared complexes : aperiodic tilings and non-residually finite groups, PhD thesis, Princeton University (1996)

[27] D T Wise, Complete square complexes, Comment. Math. Helv. 82 (2007) 683 | DOI