Uniform finite presentation for groups of polynomial growth

Abstract

We prove a quantitative refinement of the statement that groups of polynomial growth are finitely presented. Let $G$ be a group with finite generating set $S$ and let $\operatorname{Gr}(r)$ be the volume of the ball of radius $r$ in the associated Cayley graph. For each $k \geq 0$, let $R_k$ be the set of words of length at most $2^k$ in the free group $F_S$ that are equal to the identity in $G$, and let $\langle \langle R_k \rangle\rangle$ be the normal subgroup of $F_S$ generated by $R_k$, so that the quotient map $F_S/\langle\langle R_k\rangle\rangle \to G$ induces a covering map of the associated Cayley graphs that has injectivity radius at least $2^{k-1}-1$. Given a non-negative integer $k$, we say that $(G,S)$ has a new relation on scale k if $\langle\langle R_{k+1} \rangle\rangle \neq \langle\langle R_{k} \rangle\rangle$. We prove that for each $K<\infty$ there exist constants $n_0$ and $C$ depending only on $K$ and $|S|$ such that if $\operatorname{Gr}(3n)\leq K \operatorname{Gr}(n)$ for some $n\geq n_0$, then there exist at most $C$ scales $k\geq \log_2 (n)$ on which $G$ has a new relation. We apply this result in a forthcoming paper as part of our proof of Schramm's locality conjecture in percolation theory.