## Topology & Geometric Group Theory Seminar

## Fall 2008

### 1:30 – 2:30, Malott 406

Tuesday, December 9

**Misha
Ershov**, University of Virginia

*Property (T) for elementary linear groups over associative rings*

I will talk about recent joint work with Andrei Jaikin-Zapirain where
we prove that for any finitely generated (associative) ring R and any
integer n≥3 the elementary linear group EL_{n}(R) has property (T).

The majority of previous papers on property (T) for elementary linear
groups over rings were based on the bounded generation method
introduced by Shalom about 10 years ago. Using a recent
generalization of that method, also due to Shalom, Vaserstein proved
that the group EL_{n}(R), n≥3, has property (T) when R is
commutative—this generalized earlier results by Kassabov and
Nikolov and by Shalom.

Like aforementioned works, our proof uses results of Shalom and
Kassabov on relative property (T), but makes virtually no use of
bounded generation. Instead we establish a new spectral ctirerion for
property (T)—it is based on the notion of codistance between a
finite family of subgroups of a group which generalizes the notion of
epsilon-orthogonality introduced in the 2002 paper by Dymara and
Januszkiewicz "Cohomology of buildings and their automorphism
groups".

Back to seminar home page.