Abstract:
We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(\Gamma, 1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory.

Abstract:
A group G is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of G remain non-conjugate in some finite quotient of G. We prove that the free groups and the fundamental groups of finite trees of finite groups with some normalizer condition are SCS. We also introduce the subgroup into-conjugacy separability property and prove that the above groups have this property too.

Abstract:
Let $F_n$ be the free group on $n\ge 2$ elements and $\A(F_n)$ its group of automorphisms. In this paper we present a rich collection of linear representations of $\A(F_n)$ arising through the action of finite index subgroups of it on relation modules of finite quotient groups of $F_n$. We show (under certain conditions) that the images of our representations are arithmetic groups.

Abstract:
In this paper we introduce some new methods to understand the analytic behaviour of the zeta function of a group. We can then combine this knowledge with suitable Tauberian theorems to deduce results about the growth of subgroups in a nilpotent group. In order to state our results we introduce the following notation. For \alpha a real number and N a nonnegative integer, define s_N^\alpha(G) = sum_{n=1}^N a_n(G)/n^\alpha. Main Theorem: Let G be a finitely generated nilpotent infinite group. (1) The abscissa of convergence \alpha(G) of \zeta_G(s) is a rational number and \zeta_G(s) can be meromorphically continued to Re(s)>\alpha(G)-\delta for some \delta >0. The continued function is holomorphic on the line \Re(s) = (\alpha)G except for a pole at s=\alpha(G). (2) There exist a nonnegative integer b(G) and some real numbers c,c' such that s_{N}(G) ~ c N^{\alpha(G)}(\log N)^{b(G)} s_{N}^{\alpha(G)}(G) ~ c' (\log N)^{b(G)+1} for N\rightarrow \infty .

Abstract:
We start discussing the group of automorphisms of the field of complex numbers, and describe, in the special case of polynomials with only two critical values, Grothendieck's program of 'Dessins d' enfants', aiming at giving representations of the absolute Galois group. We describe Chebycheff and Belyi polynomials, and other explicit examples. As an illustration, we briefly treat difference and Schur polynomials. Then we concentrate on a higher dimensional analogue of the triangle curves, namely, Beauville surfaces and varieties isogenous to a product. We describe their moduli spaces, and show how the study of these varieties leads to new interesting questions in the theory of finite (simple) groups.

Abstract:
We show that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the set of connected components of the moduli spaces of surfaces of general type, and also that for each element $\sigma \in Gal(\bar{\Q} /\Q)$ different from the identity and from complex conjugation, there is a surface of general type such that $X$ and the Galois conjugate variety $X^{\sigma}$ have nonisomorphic fundamental groups. The result was announced by the second author at the Alghero Conference 'Topology of algebraic varieties' in september 2006. Before the present paper was actually written, we received a very interesting preprint by Robert Easton and Ravi Vakil (\cite{e-v}), where it is proven, with a completely different type of examples, that the Galois group $Gal(\bar{\Q} /\Q)$ operates faithfully on the set of irreducible components of the moduli spaces of surfaces of general type. We also give other simpler examples of surfaces with nonisomorphic fundamental groups which are Galois conjugate, hence have isomorphic algebraic fundamental groups.

Abstract:
We give a canonical procedure associating to an algebraic number a first a hyperelliptic curve C_a, and then a triangle curve (D_a, G_a) obtained through the normal closure of an associated Belyi function. In this way we show that the absolute Galois group Gal(\bar{\Q} /\Q) acts faithfully on the set of isomorphism classes of marked triangle curves, and on the set of connected components of marked moduli spaces of surfaces isogenous to a higher product (these are the free quotients of a product C_1 x C_2 of curves of respective genera g_1, g_2 >= 2 by the action of a finite group G). We show then, using again the surfaces isogenous to a product, first that it acts faithfully on the set of connected components of moduli spaces of surfaces of general type (amending an incorrect proof in a previous ArXiv version of the paper); and then, as a consequence, we obtain that for every element \sigma \in \Gal(\bar{\Q} /\Q), not in the conjugacy class of complex conjugation, there exists a surface of general type X such that X and the Galois conjugate surface X^{\sigma} have nonisomorphic fundamental groups. Using polynomials with only two critical values, we can moreover exhibit infinitely many explicit examples of such a situation.

Abstract:
Let $\Gamma$ be an irreducible lattice in $\PSL_2(\RR)^d$ ($d\in\NN$) and $z$ a point in the $d$-fold direct product of the upper half plane. We study the discrete set of componentwise distances ${\bf D}(\Gm,z)\subset \RR^d$ defined in (1). We prove asymptotic results on the number of $\gm\in\Gm$ such that $d(z,\gamma z$ is contained in strips expanding in some directions and also in expanding hypercubes. The results on the counting in expanding strips are new. The results on expanding hypercubes % improve the error terms improve the existing error terms (by Gorodnick and Nevo) and generalize the Selberg error term for $d=1$. We give an asymptotic formula for the number of lattice points $\gamma z$ such that the hyperbolic distance in each of the factors satisfies $d((\gamma z)_j, z_j)\le T$. The error term, as $T \to \infty$ generalizes the error term given by Selberg for $d=1$, also we describe how the counting function depends on $z$. We also prove asymptotic results when the distance satisfies $A_j \le d((\gamma z)_j, z_j) < B_j$, with fixed $A_j < B_j$ in some factors, while in the remaining factors $0 \le d((\gamma z)_j, z_j) \le T$ is satisfied.

Abstract:
This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H^1(G,E_n), where Gamma is a lattice in SL(2,C) and E_n is one of the standard self-dual modules. In the case Gamma = SL(2,O) for the ring of integers O in an imaginary quadratic number field, we make the theory of lifting explicit and obtain lower bounds linear in n. We have accumulated a large amount of experimental data in this case, as well as for some geometrically constructed and mostly non-arithmetic groups. The computations for SL(2,O) lead us to discover two instances with non-lifted classes in the cohomology. We also derive an upper bound of size O(n^2 / log n) for any fixed lattice Gamma in the general case. We discuss a number of new questions and conjectures suggested by our results and our experimental data.