Random quotients of the modular group are rigid and essentially incompressible

We show that for any positive integer m ≧ 1, m-relator quotients of the modular group M = PSL(2,ℤ) generically satisfy a very strong Mostow-type 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 Im (n) of 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.