Uniform finite presentation for groups of polynomial growth

TL;DR

This paper refines the understanding of groups with polynomial growth by providing a quantitative bound on the number of new relations at different scales, linking geometric growth conditions to algebraic presentations.

Contribution

It establishes a bound on the number of scales with new relations in finitely generated groups of polynomial growth, advancing the classical finite presentation theorem.

Findings

Bound on the number of scales with new relations depending only on growth and generating set size

Connection between volume growth conditions and algebraic relations in groups

Application to Schramm's locality conjecture in percolation theory

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.