Arithmeticity and totally geodesic subspaces of hyperbolic orbifolds and manifolds

The topic we will discuss is a fascinating blend of many beautiful ideas and methods from discrete subgroups of Lie groups, geometric group theory, hyperbolic geometry, geometric topology, and number theory.
In this talk I will give a survey of this beautiful domain (with all required definitions) and will present our recent results with Misha Belolipetsky (IMPA, Brazil), Sasha Kolpakov (Univ. Neuchâtel, Switzerland) and Leone Slavich (Univ. Pavia, Italy). Our main result is the following arithmeticity criterion: a hyperbolic manifold or an orbifold M is arithmetic if and only if it has infinitely many totally geodesic immersed subspaces that are so-called fc-subspaces, i.e. correspond to finite subgroups of the commensurator of a fundamental group $\Gamma = \pi_1(M).$ And M is non-arithmetic if and only if it has only finitely many such fc-subspaces. We also provide interesting examples from hyperbolic reflection groups (Vinberg’s theory), hyperbolic knot theory, and non-arithmetic hybrids constructed by Gromov and Piatetski-Shapiro.
Good references for this talk are our recent papers https://arxiv.org/abs/2002.11445(Bogachev, Kolpakov, IMRN 2021) and https://arxiv.org/abs/2105.06897
Conference ID: 942 0186 5629 Password is a six-digit number, the first three digits of which form the number p + 44, and the last three digits are the number q + 63, where p, q is the largest pair of twin primes less than 1000