Zariski density and computing in arithmetic groups

For n > 2, let Gamma _n denote either SL(n, {Z}) or Sp(n, {Z}). We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group H\leq \Gamma _n. This forms the main component of our methods for computing with such arithmetic groups H. More generally, we provide algorithms for computing with Zariski dense groups in Gamma _n. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups

Linear groups and computation

Funding: A. S. Detinko is supported by a Marie Skłodowska-Curie Individual Fellowship grant (Horizon 2020, EU Framework Programme for Research and Innovation).We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in this class of groups are surveyed. We illustrate the solution of hard mathematical problems by computer experimentation. Possible avenues for further progress are discussed.PostprintPeer reviewe

Linear groups and computation

We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for this class of groups are surveyed. We illustrate the solution of hard mathematical problems by computer experimentation. Possible avenues for further progress are discussed

Geometry of Selberg's bisectors in the symmetric space SL(n,R)/SO(n,R)SL(n,\mathbb{R})/SO(n,\mathbb{R})

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