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 $\mathrm{Gr}(r)$ be the volume of the ball of radius $r$ in the associated Cayley graph. For each $k\ge 0$, 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 $⟨⟨R_k⟩⟩$ be the normal subgroup of $F_S$ generated by $R_k$, so that the quotient map $F_S/⟨⟨R_k⟩⟩\to G$ 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 $⟨⟨R_{k+1}⟩⟩\ne ⟨⟨R_k⟩⟩$. We prove that for each $K<\infty$ there exist constants $n_0$ and $C$ depending only on $K$ and $|S|$ such that if $\mathrm{Gr}(3n)\le K\mathrm{Gr}(n)$ for some $n\ge n_0$, then there exist at most $C$ scales $k\ge {\log }_2(n)$ 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.

29 pages. Published in Discrete Analysis