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DTSTAMP:20200125T183524Z
UID:1573572600@ist.ac.at
DTSTART:20191112T163000
DTEND:20191112T173000
DESCRIPTION:Speaker: Gábor Pete\nhosted by M. Beiglboeck\, N. Berestycki\,
L. Erdoes\, J. Maas\nAbstract: A probabilistic definition of groups with
Kazhdan's property (T)\, due to Glasner & Weiss (1997)\, is that on an
y Cayley graph G of the group\, for any ergodic group-invariant random bla
ck-and-white colouring of the vertices\, with the density of each colour b
ounded away from 0\, the density of edges connecting black to white vertic
es remains bounded away from zero. Amenable groups and free groups do not
have property (T)\, while SL_d(\\Z) with d\\geq 3 do. The cost of a transi
tive graph is one half of the infimum of the expected degree of invariant
connected spanning subgraphs. Amenable transitive graphs and Cayley graphs
of SL_d(\\Z) with d\\geq 3 have cost 1\, while any Cayley graph of the fr
ee group on d generators has cost d\, by Gaboriau (2000). A question of Ga
boriau aims to connect cost with the first L^2-Betti number of groups. For
Kazhdan groups\, the latter has been known to be 0 since Bekka & Valett
e (1997)\, and Gaboriau's question then suggests that the cost of any Kaz
hdan Cayley graph should be 1. This is what we prove\, in joint work with
Tom Hutchcroft (Cambridge).
LOCATION:SR 14\, 2 OG.\, OMP 1\, University of Vienna\, IST Austria
ORGANIZER:birgit.oosthuizen-noczil@ist.ac.at
SUMMARY:Kazhdan groups have cost 1
URL:https://talks-calendar.app.ist.ac.at/events/2389
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