Journal article

### Universal lattices and Property $τ$

Abstract:

We prove that the universal lattices -- the groups $G=\SL_d(R)$ where $R=\Z[x_1,...,x_k]$, have property $\tau$ for $d\geq 3$. This provides the first example of linear groups with $\tau$ which do not come from arithmetic groups. We also give a lower bound for the expanding constant with respect to the natural generating set of $G$. Our methods are based on bounded elementary generation of the finite congruence images of $G$, a generalization of a result by Dennis and Stein on $K_2$ of some f...

Publication status:
Published

### Access Document

Publisher copy:
10.1007/s00222-005-0498-0

### Authors

More by this author
Institution:
University of Oxford
Division:
MPLS
Department:
Mathematical Institute
Role:
Author
Journal:
INVENTIONES MATHEMATICAE
Volume:
165
Issue:
1
Pages:
209-224
Publication date:
2006-07-01
DOI:
EISSN:
1432-1297
ISSN:
0020-9910
Source identifiers:
354362
Keywords:
Pubs id:
pubs:354362
UUID:
uuid:ee5d1a9e-d966-495c-9f98-e41f5977e5cc
Local pid:
pubs:354362
Deposit date:
2013-11-16