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.
Categories: math.GR math.GT