2023/08/23 by Philip Easo, Tom Hutchcroft, Easo, Philip +1 · 1 citation
Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2308.12428
openalex publication_date 2023/08/23 · openalex created_date 2023/08/26 · openalex updated_date 2026/07/28
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 Gr(r) be the volume of the ball of radius r in the associated Cayley graph. For each k ≥ 0, let Rk be the set of words of length at most 2k in the free group FS that are equal to the identity in G, and let ⟨ ⟨ Rk ⟩⟩ be the normal subgroup of FS generated by Rk, so that the quotient map FS/⟨⟨ Rk⟩⟩ → G induces a covering map of the associated Cayley graphs that has injectivity radius at least 2k-1-1. Given a non-negative integer k, we say that (G,S) has a new relation on scale k if ⟨⟨ Rk+1 ⟩⟩ ≠ ⟨⟨ Rk ⟩⟩. We prove that for each K<∞ there exist constants n0 and C depending only on K and |S| such that if Gr(3n)≤ K Gr(n) for some n≥ n0, then there exist at most C scales k≥ log2 (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.