Abstract: The purpose of this paper is to write down a complete proof of the Morrison-Kawamata cone conjecture for abelian varieties. The conjecture predicts, roughly speaking, that for a large class of varieties (including all smooth varieties with numerically trivial canonical bundle) the automorphism group acts on the nef cone with rational polyhedral fundamental domain. (See Section 1 for a precise statement.) The conjecture has been proved in dimension 2 by Sterk-Looijenga, Namikawa, Kawamata, and Totaro [Ste85, Nam85, Kaw97, Tot 10], but in higher dimensions little is known in general. Abelian varieties provide one setting in which the conjecture is tractable, because in this case the nef cone and the automorphism group can both be viewed as living inside a larger object, namely the real endomorphism algebra. In this paper we combine this fact with known results for arithmetic group actions on convex cones to produce a proof of the conjecture for abelian varieties.

Abstract: We prove a result on the singularities of ball quotients $${\Gamma\backslash\mathbb{C}{H^n}}$$ by an arithmetic group. More precisely, we show that a ball quotient has at most canonical singularities under certain restrictions on the dimension n and the underlying lattice. We also extend this result to the toroidal compactification.

Abstract: We show that the image of the pure braid group under the monodromy action on the homology of a cyclic covering of degree d of the projective line is an arithmetic group provided the number of branch points is sufficiently large compared to the degree

Abstract: A general structure theorem on higher order invariants is proven. For an arithmetic group, the structure of the corresponding Hecke module is determined. It is shown that the module does not contain any irreducible submodule. This explains the fact that L-functions of higher order forms have no Euler product. Higher order cohomology is introduced, classical results of Borel are generalized, and a higher order version of Borel’s conjecture is stated.

Abstract: We address two questions of Simon Thomas. First, we show that for any n>2 one can find a four generated free subgroup of SLn(Z) which is profinitely dense. More generally, we show that an arithmetic group \Gamma which admits the congruence subgroup property, has a profinitely dense free subgroup with an explicit bound of its rank. Next, we show that the set of profinitely dense, locally free subgroups of such an arithmetic group \Gamma is uncountable.

Abstract: Let Γ be a convex co-compact Fuchsian group. We formulate a conjecture on the critical line, i.e. what is the largest half-plane with finitely many resonances for the Laplace operator on the infinite-area hyperbolic surface \({X = \Gamma \backslash \mathbb{H}^2}\). An upper bound depending on the dimension δ of the limit set is proved which is in favor of the conjecture for small values of δ and in the case when δ > 1/2 and Γ is a subgroup of an arithmetic group. New omega lower bounds for the error term in the hyperbolic lattice point counting problem are derived.