Abstracts
I will talk about some recent results of a joint work with T. Gelander, A. Lubotzky and A. Shalev. We give estimates on the number $AL_H(x)$ of arithmetic lattices of covolume at most $x$ in the groups $H=PSL(2,R)$ and $PSL(2,C)$. Our result is especially strong for $H=PSL(2,R)$ for which we prove that $$ \lim_{x\to\infty}\frac{\log AL_H(x)}{x\log x}=\frac{1}{2\pi}. $$
In 1963 G. Baumslag proved that the full automorphism group Aut(G), of a finitely generated residually finite group G, is residually finite. In general, this result cannot be extended to the outer automorphism group Out(G)=Aut(G)/InnG. In fact, Bumagina and Wise showed that for any finitely presented group S, there exists a residually finite finitely generated group G, such that S is isomorphic to Out(G). During the talk we will discuss various assumptions on G, which give more control over Out(G). In particular, we will show that if, in addition to finite generation and residual finiteness, G has infinitely many ends, then Out(G) is residually finite.
I will report on joint work with Martin Bridson, Chuck Miller and Hamish Short. We give a complete characterisation of finitely presented residually free groups as subgroups of direct products of limit groups satisfying certain additional properties. The embedding of a given residually free group into such a direct product can be recovered algorithmically from any finite presentation, which helps us to solve certain decision problems such as the conjugacy problem and membership problem for this class of groups.