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CATEGORIES:Geometric Group Theory (GGT) Seminar
SUMMARY:Combinatorics via Closed Orbits: Vertex Expansion
and Graph Quantum Ergodicity - Amitay Kamber (Univ
ersity of Cambridge)
DTSTART;TZID=Europe/London:20210625T134500
DTEND;TZID=Europe/London:20210625T144500
UID:TALK160126AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/160126
DESCRIPTION:The symmetric space of $SL_2(R)$ is the hyperbolic
plane\, and the fact that $SL_2(Z)$ is a lattice
in $SL_2(R)$ implies that after taking a quotient
we get a finite volume hyperbolic surface. \n\nWhe
n $SL_2(R)$ is replaced by the p-adic group $SL_2(
Q_p)$ the symmetric space is a (q+1)-regular Bruha
t-Tits tree. Ihara\, Margulis and Lubotzky-Phillip
s-Sarnak observed that when $SL_2(Z)$ is replaced
by a lattice coming from a quaternion algebra one
gets a (q+1)-regular graph. Using deep number theo
retic results from the theory of automorphic forms
\, related to the classical Ramanujan conjecture\,
they showed that the resulting graphs are expande
rs with an optimal spectral gap\, i.e.\, “Ramanuja
n graphs”.\n\nThe number-theoretic Ramanujan graph
s have a lot of combinatorial applications and wer
e generalized to various combinatorial number theo
retic constructions.\n\nHowever\, there are some n
otorious open questions about those constructions\
, such as the vertex expansion of number theoretic
Ramanujan graphs.\n\nIn the talk\, I will describ
e how one can construct extermal substructures of
some number-theoretic structures\, which provides
counterexamples for many open problems. The idea i
s group-theoretic and simple - we use closed orbit
s of subgroups\, when those subgroups are availabl
e. The implementation of the idea requires some nu
mber theory.\n\nI will try to appeal to a wide aud
ience\, and focus on the group theory involved.\n\
nBased on joint work with Tali Kaufman.\n\n
LOCATION:Zoom https://maths-cam-ac-uk.zoom.us/j/95208706709
CONTACT:
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