Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


On the linearity of lattices in affine buildings and ergodicity of the singular Cartan flow
HTML articles powered by AMS MathViewer

by Uri Bader, Pierre-Emmanuel Caprace and Jean Lécureux
J. Amer. Math. Soc. 32 (2019), 491-562
Published electronically: November 27, 2018


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.
Similar Articles
Bibliographic Information
  • Uri Bader
  • Affiliation: Department of Mathematics, Weizmann Institute of Science, 7610001 Rehovot, Israel
  • MR Author ID: 707409
  • Email:
  • Pierre-Emmanuel Caprace
  • Affiliation: UCLouvain, IRMP, Chemin du Cyclotron 2, Box L7.01.02, 1348 Louvain-la-Neuve, Belgium
  • MR Author ID: 752356
  • Email:
  • Jean Lécureux
  • Affiliation: Département de Mathématiques, Bâtiment 307, Faculté des Sciences d’Orsay, Université Paris-Sud 11, F-91405 Orsay, France
  • Email:
  • 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
  • © Copyright 2018 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 32 (2019), 491-562
  • MSC (2010): Primary 20E42, 20F65, 22E40, 51E24; Secondary 22D40, 20E08, 22F50, 20C99
  • DOI:
  • MathSciNet review: 3904159