Type: Seminar

Distribution: World

Expiry: 14 Mar 2011

Auth: athomas(.pmstaff;2039.2002)@p615.pc.maths.usyd.edu.au

The next Infinite Groups Seminar will be on Monday 14 March at the University of Sydney, with Lawrence Reeves (Melbourne) and Jonathan Hillman (Sydney) speaking. The schedule including titles and abstracts is as follows. 12 noon - 1pm, Carslaw 829 (Access Grid Room): Speaker: Lawrence Reeves, University of Melbourne Title: Relatively hyperbolic groups Abstract: Relatively hyperbolic groups generalise geometrically finite groups (in the classical sense) and hyperbolic groups (in the Gromov sense). I’ll present an overview, including some recent results and open questions. 1-2:30pm: Lunch 2:30-3:30pm, Carslaw 707A: Speaker: Jonathan Hillman, University of Sydney Title: Applications of $L^2$ methods to infinite groups Abstract: A finite presentation $\mathcal{P}$ of a group $G$ determines a finite 2-complex $C(\mathcal{P})$, with Euler characteristic $\chi(C(\mathcal{P}))=1-g+r$, where $g$ and $r$ are the numbers of generators and relators of the presentation, respectively. The Euler characteristic of a finite complex is multiplicative under passage to finite covers, and is also the alternating sum of its Betti numbers. Less well known is that it is also the alternating sum of its $L^2$-Betti numbers, which are multiplicative under passage to finite covers (unlike the usual Betti numbers). We shall sketch the definition of the $L^2$-Betti numbers and show how they may be used to obtain strong results on groups with finite presentations of deficiency $g-r>0$. For instance, if $G$ is such a group and the commutator subgroup $G’$ is also finitely presentable then $G’$ is free. With further work of Kochloukova, on Novikov extensions of group rings, it suffices to assume $G’$ finitely generated.