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On the linearity of lattices in affine buildings and ergodicity of the singular Cartan flow

Authors: Uri Bader, Pierre-Emmanuel Caprace and Jean Lécureux
Journal: J. Amer. Math. Soc. 32 (2019), 491-562
MSC (2010): Primary 20E42, 20F65, 22E40, 51E24; Secondary 22D40, 20E08, 22F50, 20C99
Published electronically: November 27, 2018
MathSciNet review: 3904159
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Abstract: Let $X$ be a locally finite irreducible affine building of dimension $\geq 2$, and let $\Gamma \leq \operatorname {Aut}(X)$ be a discrete group acting cocompactly. The goal of this paper is to address the following question: When is $\Gamma$ linear? More generally, when does $\Gamma$ admit a finite-dimensional representation with infinite image over a commutative unital ring? If $X$ is the Bruhat–Tits building of a simple algebraic group over a local field and if $\Gamma$ is an arithmetic lattice, then $\Gamma$ is clearly linear. We prove that if $X$ is of type $\widetilde {A}_2$, then the converse holds. In particular, cocompact lattices in exotic $\widetilde {A}_2$-buildings are nonlinear. As an application, we obtain the first infinite family of lattices in exotic $\widetilde {A}_2$-buildings of arbitrarily large thickness, providing a partial answer to a question of W. Kantor from 1986. We also show that if $X$ is Bruhat–Tits of arbitrary type, then the linearity of $\Gamma$ implies that $\Gamma$ is virtually contained in the linear part of the automorphism group of $X$; in particular, $\Gamma$ is an arithmetic lattice. The proofs are based on the machinery of algebraic representations of ergodic systems recently developed by U. Bader and A. Furman. The implementation of that tool in the present context requires the geometric construction of a suitable ergodic $\Gamma$-space attached to the the building $X$, which we call the singular Cartan flow.

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Additional Information

Uri Bader
Affiliation: Department of Mathematics, Weizmann Institute of Science, 7610001 Rehovot, Israel
MR Author ID: 707409

Pierre-Emmanuel Caprace
Affiliation: UCLouvain, IRMP, Chemin du Cyclotron 2, Box L7.01.02, 1348 Louvain-la-Neuve, Belgium
MR Author ID: 752356

Jean Lécureux
Affiliation: Département de Mathématiques, Bâtiment 307, Faculté des Sciences d’Orsay, Université Paris-Sud 11, F-91405 Orsay, France

Received by editor(s): October 7, 2016
Received by editor(s) in revised form: July 2, 2018, August 20, 2018, and September 18, 2018
Published electronically: November 27, 2018
Additional Notes: The first author acknowledges the support of ERC grant #306706.
The second author ackowledges the support of F.R.S.-FNRS and of ERC grant #278469.
The third author was supported in part by ANR grant ANR-14-CE25-0004 GAMME and ANR-16-CE40-0022-01 AGIRA
Article copyright: © Copyright 2018 American Mathematical Society