a. Overview of results of the project
a.1 Methodology. We developed novel techniques for computing with finitely generated linear groups over infinite domains based on methods for linear algebraic groups and Lie algebras,
and results for discrete subgroups of Lie groups.
New methods and algorithms for practical computing with matrix groups over finite rings are of independent interest.
a.2 Algorithms. The following algorithms designed in the project are milestones: testing Zariski density; computing congruence quotients of dense subgroups (computer realization of the strong
approximation theorem); computing arithmetic closure of dense subgroups of semisimple algebraic groups satisfying congruence subgroup property; algorithms for linear groups of finite rank over number fields.
a.3 Software and applications. Our algorithms have been implemented in computer algebra systems GAP, Magma.
The software designed in the project was used to investigate properties and structure of groups which arise in applications of group theory in a number of areas.
b. Exploitation and dissemination of results.
b.1 The results obtained in the project available as peer-reviewed papers, preprints, and conference proceedings
(eight published or accepted for publication, and three further to be submitted in 2018).
Open access to publications provided via online repositories.
b.2 The dissemination activities include participation in eight international conferences and workshops, and ten talks delivered at conferences and research seminars world-wide.
b.3 As a part of dissemination activities we visited twelve universities and research centers around the world; supported in part by Oberwolfach Research Institute for Mathematics
(Germany, Research-in-Pairs scheme) and International Center for Mathematical Sciences (UK, Research-in-Groups scheme).
