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A survey on graphs with polynomial growth
AbstractIn this paper we give an overview on connected locally finite transitive graphs with polynomial growth. We present results concerning the following topics: •Automorphism groups of graphs with polynomial growth.•Groups and graphs with linear growth.•S-transitivity.•Covering graphs.•Automorphism groups as topological groups
Coarse geometry of the fire retaining property and group splittings
Given a non-decreasing function we
define a single player game on (infinite) connected graphs that we call fire
retaining. If a graph admits a winning strategy for any initial
configuration (initial fire) then we say that has the -retaining
property; in this case if is a polynomial of degree , we say that
has the polynomial retaining property of degree .
We prove that having the polynomial retaining property of degree is a
quasi-isometry invariant in the class of uniformly locally finite connected
graphs. Henceforth, the retaining property defines a quasi-isometric invariant
of finitely generated groups. We prove that if a finitely generated group
splits over a quasi-isometrically embedded subgroup of polynomial growth of
degree , then has polynomial retaining property of degree . Some
connections to other work on quasi-isometry invariants of finitely generated
groups are discussed and some questions are raised.Comment: 16 pages, 1 figur
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