Random quotients of the modular group are rigid and essentially incompressible

Details Random quotients of the modular group are rigid and essentially incompressible Random quotients of the modular group are rigid and essentially incompressible

2006-04-14

arxiv.org/abs/math/0604343v1

Journal f\"ur die reine und angewandte Mathematik (Crelle's Journal), 628 (2009), pp. 91--119

Mathematics

Group Theory

Scientific paper

10.1515/CRELLE.2009.019

We show that for any positive integer $m\ge 1$, $m$-relator quotients of the modular group $M = PSL(2,\mathbb{Z})$ generically satisfy a very strong Mostow-type \emph{isomorphism rigidity}. We also prove that such quotients are generically "essentially incompressible". By this we mean that their "absolute $T$-invariant", measuring the smallest size of any possible finite presentation of the group, is bounded below by a function which is almost linear in terms of the length of the given presentation. We 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 obtain other algebraic results and show that such quotients are complete, Hopfian, co-Hopfian, one-ended, word-hyperbolic groups.

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