BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Classifying group actions on hyperbolic spaces - Denis Osin (Vande
 rbilt University)
DTSTART:20250904T130000Z
DTEND:20250904T140000Z
UID:TALK233659@talks.cam.ac.uk
DESCRIPTION:For a given group G\, it is natural to ask whether one can cla
 ssify all isometric G-actions on Gromov hyperbolic spaces. I will discuss 
 a formalization of this problem based on the complexity theory of Borel eq
 uivalence relations. Our focus will be on actions of general type\, i.e.\,
  non-elementary actions without fixed points at infinity\, as these are pa
 rticularly useful from the perspective of geometric group theory. The main
  result in this direction is the following dichotomy: for every countable 
 group G\, either all general type G-actions on hyperbolic spaces can be cl
 assified by an explicit invariant ranging in an infinitely dimensional pro
 jective space\, or they are unclassifiable in a very strong sense. In term
 s of Borel complexity theory\, we show that the equivalence relation assoc
 iated with the classification problem is either smooth or K_&sigma\;-compl
 ete. Groups SL_2(F)\, where F is a countable field of characteristic 0\, s
 atisfy the former alternative\, while non-elementary hyperbolic (and\, mor
 e generally\, acylindrically hyperbolic) groups satisfy the latter. The pr
 oof of the main theorem draws on several results of independent interest a
 nd provides new insights into the boundary dynamics of group actions on hy
 perbolic spaces. The talk is based on a joint paper with K. Oyakawa.&nbsp\
 ;
LOCATION:Seminar Room 1\, Newton Institute
END:VEVENT
END:VCALENDAR