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)

# 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!

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