Strong and uniform boundedness of groups

A group [Formula: see text] is called bounded if every conjugation-invariant norm on [Formula: see text] has finite diameter. We introduce various strengthenings of this property and investigate them in several classes of groups including semisimple Lie groups, arithmetic groups and linear algebraic groups. We provide applications to Hamiltonian dynamics.

Counting and equidistribution in quaternionic Heisenberg groups

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

Arithmetic Subgroups and Applications

Arithmetic subgroups are an important source of discrete groups acting freely on manifolds. We need to know that there exist many torsion-free 푺푺L(ퟐퟐ,ℝ) is an “arithmetic” subgroup of 푺푺L(ퟐퟐ,ℝ). The other arithmetic subgroups are not as obvious, but they can be constructed by using quaternion algebras. Replacing the quaternion algebras with larger division algebras yields many arithmetic subgroups of 푺푺L(풏풏,ℝ), with 풏풏>2. In fact, a calculation of group cohomology shows that the only other way to construct arithmetic subgroups of 푺푺L(풏풏, ℝ) is by using arithmetic groups. In this paper justifies Commensurable groups, and some definitions and examples,ℝ-forms of classical simple groups over ℂ, calculating the complexification of each classical group, Applications to manifolds. Let us start with 푺푺푺푺(푛푛,ℂ). This is already a complex Lie group, but we can think of it as a real Lie group of twice the dimension. As such, it has a complexification.

Profinite invariants of arithmetic groups

We prove that the sign of the Euler characteristic of arithmetic groups with the congruence subgroup property is determined by the profinite completion. In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type F. Our methods imply similar results for $\ell^2$ -torsion as well as a strong profiniteness statement for Novikov–Shubin invariants.

Introduction

This introductory chapter presents the general principle that the cohomology of arithmetic groups and the L-functions L(s, π‎, r) attached to irreducible “pieces” π‎ have a strong symbiotic relationship with each other. The symbiosis goes in both directions. The first is that expressions in the special values L(k, π‎, r) enter in the transcendental description of the cohomology. Since the cohomology is defined over ℚ one can deduce rationality (algebraicity) results for these expressions in special values. Next, these special values in turn influence the structure of the cohomology as a Hecke module; prime numbers dividing these values occur in the denominators of Eisenstein classes.