Let (Formula presented.) be a non-elementary action by isometries of a hyperbolic group G on a (not necessarily proper) hyperbolic metric space X. We show that the set of elements of G which act as loxodromic isometries of X has density one in the word metric on G. That is, for any finite generating set of G, the proportion of elements in G of word length at most n, which are X–loxodromics, approaches 1 as n → ∞. We also establish several results about the behavior in X of the images of typical geodesic rays in G; for example, we prove that they make linear progress in X and converge to the Gromov boundary ∂X. We discuss various applications, in particular to mapping class groups, Out(FN), and right–angled Artin groups.
Counting loxodromics for hyperbolic actions / Gekhtman, I.; Taylor, S. J.; Tiozzo, G.. - In: JOURNAL OF TOPOLOGY. - ISSN 1753-8416. - 11:2(2018), pp. 379-419. [10.1112/topo.12053]
Counting loxodromics for hyperbolic actions
Tiozzo G.
2018
Abstract
Let (Formula presented.) be a non-elementary action by isometries of a hyperbolic group G on a (not necessarily proper) hyperbolic metric space X. We show that the set of elements of G which act as loxodromic isometries of X has density one in the word metric on G. That is, for any finite generating set of G, the proportion of elements in G of word length at most n, which are X–loxodromics, approaches 1 as n → ∞. We also establish several results about the behavior in X of the images of typical geodesic rays in G; for example, we prove that they make linear progress in X and converge to the Gromov boundary ∂X. We discuss various applications, in particular to mapping class groups, Out(FN), and right–angled Artin groups.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
