[Binghamton Geometry/Topology Seminar] Geometry/Topology Seminar

[Christoforos Neofytidis]

BINGHAMTON GEOMETRY/TOPOLOGY SEMINAR

   Date: Thursday, May 4th, 2017

   Time: 3.30 pm (NOTE SPECIAL TIME)

   Place: Whitney Hall, Room 100E

   Speaker: Ilya Gekhtman (Yale)

Title: Word length asymptotics for actions of some automatic (e.g.
relatively hyperbolic) groups.

Abstract: Consider any nonelementary action of a hyperbolic group G on a
not necessarily proper Gromov hyperbolic space X. The action is not
assumed to be discrete (for example, it could be a dense subgroup of
SL_{2}\R) and X is not assumed to be proper (for example it could be the
curve complex, on which the mapping class group acts with pseudo-Anosov
elements acting as loxodromics).

We prove certain asymptotic properties for the action, including the
following. 1)With respect to the Patterson-Sullivan measure on the
boundary of G, the image in X of almost every word-geodesic in G
sublinearly tracks a geodesic in X. 2)The proportion of elements in a
Cayley-ball of radius R in G which act loxodromically on X converges to 1
with R.

A major tool is Cannon's theorem that hyperbolic groups admit geodesic
automation. The same result hold for relatively hyperbolic groups with
respect to generating sets which admit a geodesic automaton, including
geometrically finite Kleinian groups, and more generally to automatic
structures satisfying certain axioms related to growth tightness. We also
obtain results for more general Markov processes, for example showing a
*nonbacktracking* random walk on a group acting nonelementarily on a
Gromov hyperbolic space hits loxodromic elements. This is based on
completed and ongoing work with Sam Taylor and Giulio Tiozzo.

NOTE:  The seminar has a webpage where the semester's program is listed:

http://www2.math.binghamton.edu/p/seminars/topsem