Raum M911

We show that for any positive integer $m/ge 1$, $m$-relator quotients of the modular group $M = PSL(2,Z)=< a,b| a^2=b^3=1>$ generically satisfy a very strong Mostow-type ``isomorphism rigidity``. In particular, we prove if two such quotients are isomorphic then their Cayley graphs on the /emph{given} generating set $a,b$ are isometric. This allows us to compute the precise asymptotics of the number $I_m(n)$ of /emph{isomorphism types} of $m$-relator quotients of $M$ where all the defining relators are cyclically reduced words of length $n$ in $M$. We also prove that random quotients of $M$ are ``essentially algebraically incompressible``, that is, they do not admit a finite group presentation of length much shorter than the given one. This talk is based on joint work with Paul Schupp.

Ilya Kapovich

UIUC Department of Mathematics