Abstract
We give a nonstandard treatment of the notion of ends of proper geodesic metric spaces. We then apply this nonstandard treatment to Cayley graphs of finitely generated groups and give nonstandard proofs of many of the fundamental results concerning ends of groups. We end with an analogous nonstandard treatment of the ends of relatively Cayley graphs, that is Cayley graphs of cosets of finitely generated groups. 2000 Mathematics Subject Classification 20F65 (primary); 26E35, 57M07 (sec- ondary)
Highlights
Nonstandard analysis made its first serious impact on geometric group theory via the work of van den Dries and Wilkie [7] on Gromov’s theorem on polynomial growth
We treat the notion of ends of a finitely generated group from a nonstandard perspective
An analysis of the ends of the Cayley graph of a finitely generated group yields a significant amount of algebraic information about the group
Summary
Nonstandard analysis made its first serious impact on geometric group theory via the work of van den Dries and Wilkie [7] on Gromov’s theorem on polynomial growth. We discuss a nonstandard property that a finitely generated group can possess, namely that the group have multiplicative ends; see Section 6. This notion suggests itself immediately once the nonstandard framework is developed, begging the question of the standard counterpart of the notion. We will need the following basic nonstandard criteria for compactness due to Robinson: X is compact if and only if Xns = X∗ (see Davis [6] for a proof) This characterization holds more generally for hausdorff topological spaces (X, τ ), where, for x ∈ X , we set μ(x) := {U∗ | U ∈ τ, x ∈ U}. N, and N , sometimes subscripted, range over N := {0, 1, 2, . . .}
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