UC Davis

We study several problems about $\mathcal{P}(n)$, the symmetric space associated with the real Lie group $SL(n,\mathbb{R})$. We endow the symmetric space $\mathcal{P}(n)$ with an $SL(n,\mathbb{R})$-invariant premetric proposed by Selberg as a substitute for the Riemannian distance. The problems addressed in this study are linked to an algorithm designed to determine generalized geometric finiteness for subgroups of $SL(n,\mathbb{R})$, similar to the algorithm proposed by Riley in hyperbolic spaces based on Poincar e's fundamental polyhedron theorem.

The main results of this dissertation are twofold. The first part consists of Chapters \ref{chp:3}-\ref{chp:4}, focusing on the ridge-cycle condition in Poincar e's fundamental polyhedron theorem. This condition requires us to determine whether given hyperplanes in $\mathcal{P}(n)$ are disjoint. We establish several criteria for the disjointness of hyperplanes in $\mathcal{P}(n)$ and construct an angle-like function between hyperplanes.

The second part, spanning Chapters \ref{chp:5} to \ref{chp:7}, concerns the proposed Poincar e's algorithm for $SL(n,\mathbb{R})$. We describe and implement an algorithm that computes the face-poset structure of Dirichlet-Selberg domains for finite subsets of $SL(n,\mathbb{R})$. This constitutes a crucial aspect of the proposed Poincar e's algorithm. Notably, Poincar e's algorithm for a given subgroup will not terminate if the subgroup lacks a finitely-sided Dirichlet-Selberg domain. This observation motivates us to categorize the Abelian subgroups of $SL(3,\mathbb{R})$ based on whether their Dirichlet-Selberg domains are finitely-sided or not.