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Amir, Gideon; Baldasso, Rangel; Gerasimova, Maria; Kozma, Gady
Discrete mathematics, January 2023, 2023-01-00, Volume: 346, Issue: 1Journal Article
We study the fire-retaining problem on groups, a quasi-isometry invariant11See Section 2 for some caveats. introduced by Martínez-Pedroza and Prytuła 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(nd−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≥1, groups that satisfy {nd}-retainment but not o(nd)-retainment, as well as groups that do not satisfy sub-exponential retainment.
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