Small Cancellation Stability and Isomorphism Rigidity for Generic Finitely Presented Groups

Abstract

Let $F_m=F(a_1,\dots ,a_m)$ with $m\ge 2$, and fix $q\ge 1$. For every fixed $0<\lambda <1$, we prove that a $q$-tuple ${\mathbf{W}}_n$ of independent uniformly random cyclically reduced words of length $n$ is \emph{$\lambda$-stable} with probability converging to $1$ exponentially fast. Namely, for every $\Phi \in Aut(F_m)$, the tuple $\Phi ({\mathbf{W}}_n)$, after cyclic reduction and symmetrization, satisfies the $C^{\prime }(\lambda )$ small cancellation condition. Combining generic $\lambda$-stability with Greendlinger normal-closure rigidity and with previous results of Kapovich-Schupp-Shpilrain on generic Nielsen uniqueness and generic Whitehead rigidity we establish, for any fixed $m\ge 2,q\ge 1$, isomorphism rigidity for generic $m$-generator $q$-relator groups. Thus we show that two such generic groups $⟨a_1,\dots ,a_m|r_1,\dots ,r_q⟩$ and $⟨a_1,\dots ,a_m|s_1,\dots ,s_q⟩$ are isomorphic if and only if, after possibly permuting and inverting the generators $a_1,\dots ,a_m$, the relator tuples $(r_1,\dots ,r_q)$ and $(s_1,\dots ,s_q)$ are the same, up to reordering, cyclic permutations and inverting the relators. Among the applications, we obtain a quadratic-time algorithm that generically solves the isomorphism problem for $m$-generator $q$-relator groups, and show that the number of isomorphism types represented by $m$-generator $q$-relator presentations with cyclically reduced relators of length $n$ is asymptotic to \[ \frac{(2m-1)^{qn}}{2^{m+q}m!\,q!\,n^q}. \] The proof of generic $\lambda$-stability relies on the use of geodesic currents and on our deterministic sufficient criterion for a $q$-tuple $\mathbb{W}$ in $F_m$ to be $\lambda$-stable in terms of the components of $\mathbb{W}$ being sufficiently projectively close to filling currents.