Infinite Groups Seminar: Hillman, Reeves

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.