Fire retainment on Cayley graphs

TL;DR

This paper investigates the fire-retaining problem on Cayley graphs, establishing bounds related to polynomial and exponential growth, and constructing groups with diverse retention behaviors.

Contribution

It proves bounds for polynomial growth Cayley graphs and constructs groups with varied retention properties in exponential growth regimes.

Findings

Polynomial growth Cayley graphs do not satisfy certain retention functions.

General lower bounds are established for exponential growth groups.

Constructed groups exhibit diverse retention behaviors, including polynomial and non-subexponential retention.

Abstract

We study the fire-retaining problem on groups, a quasi-isometry invariant introduced by Mart\'inez-Pedroza and Prytula [8], related to the firefighter problem. We prove that any Cayley graph with degree-$d$ polynomial growth does not satisfy $\{f(n)\}$-retainment, for any $f(n) = o(n^{d-2})$, matching the upper bound given for the firefighter problem for these graphs. In the exponential growth regime we prove general lower bounds for direct products and wreath products. These bounds are tight, and show that for exponential-growth groups a wide variety of behaviors is possible. In particular, we construct, for any $d\geq 1$, groups that satisfy $\{n^{d}\}$-retainment but not $o(n^d)$-retainment, as well as groups that do not satisfy sub-exponential retainment.