Faculty of Mathematics
Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics

# Project C8

## Finiteness Properties of Infinite Discrete Groups

Principal Investigator(s) Other Investigators

## Summary:

The main goal is to establish finiteness properties (finite generation, finte presentability, etc.) of arithmetic groups in positive characteristic, e.g., $SL_N(\mathbf{F}_q[t; t^{-1}])$. In particular, one aim is a proof of the {\em Rank Conjecture} which would give the exact finiteness length for reductive groups. Another point on the agenda is the question whether $SL_2(\mathbf{Z}[t; t^{-1}])$ is finitely generated. Finally the finiteness properties of Torelli subgroups (a) in Out($F_{N}$) and (b) in mappingclass groups of closed oriented surfaces are to be determined.

## Recent Preprints:

|       |                                                                                                          |             |
|-------|----------------------------------------------------------------------------------------------------------|-------------|
| 13030 | Polytopes and groups                                                                                     | PDF | PS.GZ |
| 12146 | The braided Thompson’s groups are of type $F_\infty$                                                     | PDF | PS.GZ |
| 12145 | Higher generation for pure braid groups                                                                  | PDF | PS.GZ |
| 12070 | The Brin-Thompson groups $sV$ are of type $F_\infty$                                                     | PDF | PS.GZ |
| 12069 | An Eilenberg–Ganea phenomenon for actions with virtually cyclic stabilisers                              | PDF | PS.GZ |
| 12060 | Higher Finiteness Properties of Reductive Arithmetic Groups in Positive Characteristic: The Rank Theorem | PDF | PS.GZ |
| 12055 | Rational homological stability for groups of partially symmetric automorphisms of free groups            | PDF | PS.GZ |
| 12054 | A combinatorial proof of the degree theorem in Auter space                                               | PDF | PS.GZ |
| 12053 | Brown's criterion in Bredon homology                                                                     | PDF | PS.GZ |
| 12004 | On the classifying space for the family of virtually cyclic subgroups for elementary amenable groups     | PDF | PS.GZ |

## Announced Talks interacting with the project:

May 21, 2013 16:15U2-135  !!! CANCELLED !!!Endlichkeitseigenschaften arithmetischer Gruppen über Funktionenkörpern (Teil 2)Kai-Uwe Bux
May 22, 2013 14:15T2-228  Classifying spaces and cohomology rings for Thompson's groups F and braided FMatthew Zaremsky
May 23, 2013 12:15V4-116  Ping-pong on a euclidean buildingStefan Witzel
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