On semisimple representations of universal lattices.
type: Item
instance of: scholarly article
title: On semisimple representations of universal lattices.
zbMATH Open document ID: 1197.20039
DOI: 10.4171/GGD/79
author: Daniel K. Shenfeld
published in: Groups, Geometry, and Dynamics
publication date: 2010-01-14T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_abstract.php
review text: Finite-dimensional semisimple complex representations of the universal lattices \(\Gamma_{n,k}=\text{SL}_n(\mathbb{Z}[x_1,\dots,x_k])\) (\(n\geq 3\)) are studied. One may obtain such a representation by specializing \(x_1,\dots,x_k\) to some complex values and composing the induced homomorphism \(\Gamma_{n,k}\to\text{SL}_n(\mathbb{C})\) with a rational representation of \(\text{SL}_n(\mathbb{C})\). It is shown that every semisimple representation coincides, on a subgroup of finite index, with a direct sum of tensor products of representations obtained in this way.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20G35
zbMATH DE Number: 5658123
zbMATH Keywords: universal lattices
zbMATH Keywords: superrigidity
zbMATH Keywords: arithmetic groups
zbMATH Keywords: arithmetic lattices
zbMATH Keywords: semisimple complex representations
zbMATH Keywords: tensor products of representations
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
cites work: Q5563439
cites work: \(K\)-theory and stable algebra
cites work: Solution of the congruence subgroup problem for \(\text{SL}_ n\) \((n\geq 3)\) and \(\text{Sp}_{2n}\) \((n\geq 2)\)
cites work: Q3726220
cites work: On the Structure and Ideal Theory of Complete Local Rings
cites work: Q4264394
cites work: Q4002278
cites work: On linearity of finitely generated \(R\)-analytic groups.
cites work: Universal lattices and property \(\tau\)
cites work: Connection of the dual space of a group with the structure of its closed subgroups
cites work: Q4138813
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: On the congruence subgroup problem
cites work: Le problème des groupes de congruence pour \(SL_2\)
cites work: Algèbre locale. Multiplicités. Cours au Collège de France, 1957-1958, rédigé par Pierre Gabriel. 2e ed.
cites work: Lie algebras and Lie groups. 1964 lectures, given at Harvard University.
cites work: Bounded generation and Kazhdan's property (T)
cites work: Q5490950
cites work: Linear algebraic groups.
cites work: ON THE STRUCTURE OF THE SPECIAL LINEAR GROUP OVER POLYNOMIAL RINGS
cites work: Q5566639
cites work: Q4083539
cites work: Q3687381
description: scientific article; zbMATH DE number 5658123