Colloquium: Uri Bader (Weizmann) - Totally geodesic subspaces and arithemeticity phenomena in hyperbolic manifolds

(openning talk for the 2021 Joram Seminar)

Live broadcast  \

Title: Totally geodesic subspaces and arithemeticity phenomena in hyperbolic manifolds
Abstract: In this colloquium talk I will survey in a colloquial manner a
well known, still wonderful, connection between geometry and arithmetics
and discuss old and new results in this topic. The starting point of the
story is Cartan's discovery of the correspondence between semisimple Lie
groups and symmetric spaces. Borel and Harish-Chandra, following Siegel,
later realized a fantastic further relation between arithmetic subgroups
of semisimple Lie groups and locally symmetric space - every arithmetic
group gives a locally symmetric space of finite volume. The best known
example is the modular curve which is associated in this way with the
group SL_2(Z). This relation has a partial converse, going under the
name "Arithmeticity Theorem", which was proven, under a higher rank
assumption, by Margulis and in some rank one situations by Corlette and
Gromov-Schoen. The rank one setting is related to hyperbolic geometry -
real, complex, quaternionic or octanionic. There are several open
questions regarding arithmeticity of hyperbolic manifolds of finite
volume over the real or complex fields and there are empirical evidences
relating these questions to the geometry of totally geodesic
submanifolds. Recently, some of these questions were solved by
Margulis-Mohammadi (real hyp. 3-dim), Baldi-Ullmo (complex hyp.) and
B-Fisher-Miller-Stover. The techniques involve a mixture of ergodic
theory, algebraic groups theory and Hodge theory. Details of these
techniques and proofs will be given in the coming lectures.


Thu, 17/06/2021 - 14:30 to 15:30