Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Poincaré's theorem determines if a given polyhedron in the hyperbolic space has some specific nice group structure. While the theorem also applies to $P_1(n)$, the space of positive definite matrices, it is still unclear how to verify the two conditions in the theorem's statement, namely the properness and the completeness conditions.
In $P_1(n)$, the usual Riemannian distance is sometimes replaced by Selberg's invariant. In the sense of Selberg's invariant, the sides of Dirichlet domains or segment bisectors are nice as hyperplanes in $P_1(n)$.
Our recent work defined an angle function between Selberg's segment bisectors, which provides a way to verify the properness condition. We also found a criterion to determine if two bisectors intersect in $P_1(n)$.