A+ CATEGORY SCIENTIFIC UNIT

Coarse structures and group actions

Volume 111 / 2008

N. Brodskiy, J. Dydak, A. Mitra Colloquium Mathematicum 111 (2008), 149-158 MSC: Primary 54F45, 54C55; Secondary 54E35, 18B30, 54D35, 54D40, 20H15 DOI: 10.4064/cm111-1-13

Abstract

The main results of the paper are:

Proposition 0.1. A group $G$ acting coarsely on a coarse space $(X,{\mathcal{C}})$ induces a coarse equivalence $g\mapsto g\cdot x_0$ from $G$ to $X$ for any $x_0\in X$.

Theorem 0.2. Two coarse structures ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ on the same set $X$ are equivalent if the following conditions are satisfied:

(1) Bounded sets in ${\mathcal{C}}_1$ are identical with bounded sets in ${\mathcal{C}}_2$.

(2) There is a coarse action $\phi_1$ of a group $G_1$ on $(X,{\mathcal{C}}_1)$ and a coarse action $\phi_2$ of a group $G_2$ on $(X,{\mathcal{C}}_2)$ such that $\phi_1$ commutes with $\phi_2$.

They generalize the following two basic results of coarse geometry:

Proposition 0.3 (Shvarts–Milnor lemma [5, Theorem 1.18]). A group $G$ acting properly and cocompactly via isometries on a length space $X$ is finitely generated and induces a quasi-isometry equivalence $g\mapsto g\cdot x_0$ from $G$ to $X$ for any $x_0\in X$.

Theorem 0.4 (Gromov [4, p. 6]). Two finitely generated groups $G$ and $H$ are quasi-isometric if and only if there is a locally compact space $X$ admitting proper and cocompact actions of both $G$ and $H$ that commute.

Authors

  • N. BrodskiyDepartment of Mathematics
    University of Tennessee
    Knoxville, TN 37996, U.S.A.
    e-mail
  • J. DydakDepartment of Mathematics
    University of Tennessee
    Knoxville, TN 37996, U.S.A.
    e-mail
  • A. MitraDepartment of Mathematics
    University of Tennessee
    Knoxville, TN 37996, U.S.A.
    e-mail

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