### A Note on the Discontinuous Subgroups of the Orthogonal Group.

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We establish the spectral gap property for dense subgroups of SU$\left(d\right)$$(d\ge 2)$, generated by finitely many elements with algebraic entries; this result was announced...

There are errors in the proof of uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture, and we give several remarks on further developments.

We give necessary and sufficient conditions for an orthogonal group defined over a global field of characteristic $\ne 2$ to contain a maximal torus of a given type.

We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.

We provide partial results towards a conjectural generalization of a theorem of Lubotzky-Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric inequalities and finiteness properties. As a tool in our proof, we establish polynomial isoperimetric inequalities and finiteness properties for certain solvable groups that appear as subgroups of parabolic groups in semisimple groups, thus generalizing a theorem of Bux....