**SMS scnews item created by Stephan Tillmann at Wed 6 Aug 2014 1625**

Type: Seminar

Distribution: World

Expiry: 5 Nov 2014

**Calendar1: 13 Aug 2014 1105-1155**

**CalLoc1: Carslaw 535A**

CalTitle1: Geometry-Topology-Analysis Seminar: Abramenko -- On finite and elementary generation of \(SL_2(R)\)

Auth: tillmann@p710.pc (assumed)

### Geometry-Topology-Analysis Seminar

# On finite and elementary generation of \(SL_2(R)\)

### Peter Abramenko

GTA Seminar in Week 3 -- Wednesday, 13 August, 11:00-12:00 in Carslaw 535A

Speaker: **Peter Abramenko (Virginia) **

Title: **On finite and elementary generation of \(SL_2(R)\) **

Abstract: Let \(R\) be an integral domain which is finitely generated as a ring.
Interesting questions regarding \(SL_2(R)\) are whether this group is finitely generated
or whether it is generated by elementary matrices. I will explain how these two
questions are related, and present a brief survey of some well-known results in this
context.

The main part of the talk will be devoted to the following (new) Theorem: Let
\(R_0\) be a finitely generated integral domain of Krull dimension greater than 1 (e.g.
\(R_0 = Z[x]\) or \(F_q[x,y]),\) \(R = R_0[t]\) the polynomial ring over \(R_0\) and
\(F\) the field of fractions of \(R.\) Then *no* subgroup of \(SL_2(F)\) containing
\(SL_2(R)\) is finitely generated.

I will also explain why the action of \(SL_2(F)\) on
an appropriate (Bruhat-Tits) tree is an important ingredient of the proof of this
theorem.

Please join us for lunch after the talk!