Continuously many quasiisometry classes of \(2\)-generator groups (Q1392745)

Summary: In the course of his construction of groups of intermediate growth, Grigorchuk showed that there are continuously many quasiisometry classes of 2-generator groups. In this paper, we describe another class of groups exhibiting the latter phenomenon, and for which the demonstration is elementary. Unlike those of Grigorchuk, our groups have exponential growth, and can be taken to be torsion free. In fact, they can be exhibited explicitly as small cancellation groups as follows. Let \({\mathcal P}(\mathbb{N})\) be the set of subsets of the natural numbers, \(\mathbb{N}\). Given \(F,F'\in{\mathcal P}(\mathbb{N})\), we write \(F\sim F'\) if the symmetric difference of \(F\) and \(F'\) is finite. This defines an equivalence relation on \({\mathcal P}(\mathbb{N})\) with every equivalence class countable. There are thus continuously many equivalence classes. Given \(F\in{\mathcal P}(\mathbb{N})\), let \(S(F)=\{2^{2^n}\mid n\in F\}\). Given \(p\in\mathbb{N}\), let \(w_p(a,b)=(a^pb^p)^7\) be the (cyclic) word in two letters, \(a\) and \(b\). Given \(S\subseteq\mathbb{N}\), let \(\Gamma(S)\) be the group with presentation \(\langle a,b\mid (w_p(a,b))_{p\in S}\rangle\). We show Proposition 1. If \(F,F'\in{\mathcal P}(\mathbb{N})\) are such that \(\Gamma(S(F))\) and \(\Gamma(S(F'))\) are quasiisometric, then \(F\sim F'\).