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SUMMARY:Some uniformly bounded representations of hyperbolic groups - Jan 
 Spakula (University of Southampton)
DTSTART:20251007T143000Z
DTEND:20251007T153000Z
UID:TALK236236@talks.cam.ac.uk
DESCRIPTION:We prove that some of the boundary representations of (Gromov)
  hyperbolic groups are uniformly bounded.\nOne can construct complementary
  series representations of SL(2\,R) from its action on the circle\; this w
 ork is an attempt to generalise parts of this theory to hyperbolic groups.
 \nMore concretely: Suppose G is a hyperbolic group\, acting geometrically 
 on a (strongly) hyperbolic space X. For this talk\, "boundary representati
 ons" are linear representations &pi\;z of G coming from the action of G on
  the Gromov boundary Z of X. These are parametrised by a complex parameter
  z with 0<Re(z)<1. For z=1/2\, &pi\;z is the (unitary) quasi-regular repre
 sentation on L&sup2\;(Z). For Re(z)&ne\;1/2\, there is no obvious unitary 
 structure for &pi\;z.\nDenote by D the Hausdorff dimension of Z. For 1/2 -
  1/D < Re(z) < 1/2\, we construct function spaces on Z on which &pi\;z bec
 ome uniformly bounded.\nThis is joint work with Kevin Boucher.
LOCATION:Seminar Room 1\, Newton Institute
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