Abstract Let $X \subset \mathbb{C}^{n}$ be an algebraic variety, and let $\Lambda a discrete subgroup whose real complex spans agree. We describe the topological closure of image $X$ in $\mathbb{C}^{n} / \Lambda $, thereby extending result Peterzil–Starchenko case when $ is cocompact. also obtain similar extension $X\subset \mathbb{R}^{n}$ definable o-minimal structure with no restrictions on a...

We present a topological proof of the following theorem of Benoist-Quint: for a finitely generated non-elementary discrete subgroup Γ1 of PSL(2,R) with no parabolics, and for a cocompact lattice Γ2 of PSL(2,R), any Γ1 orbit on Γ2\PSL(2,R) is either finite or dense.

We show that a number of different notions of dimension coincide for length spaces with curvature bounded above. We then apply this result, showing that if X is a locally compact CAT (0) space with cocompact isometry group, then the dimension of the Tits boundary and the asymptotic cone(s) of X are determined by the maximal dimension of a flat in X .

we describe under various conditions abelian subgroups of the automorphism‎ ‎group $mathrm{aut}(t_{n})$ of the regular $n$-ary tree $t_{n}$‎, ‎which are‎ ‎normalized by the $n$-ary adding machine $tau =(e‎, ‎dots‎, ‎e,tau )sigma _{tau‎ ‎}$ where $sigma _{tau }$ is the $n$-cycle $left( 0,1‎, ‎dots‎, ‎n-1right) $‎. ‎as‎ ‎an application‎, ‎for $n=p$ a prime number‎, ‎and for $n=4$‎, ‎we prove that...

If a group acts by uniformly quasi-Möbius homeomorphisms on a compact Ahlfors n-regular space of topological dimension n such that the induced action on the space of distinct triples is cocompact, then the action is quasi-symmetrically conjugate to an action on the standard n-sphere by Möbius transformations.

A twisted torus is a nilmanifold which is the quotient of a real Heisenberg group by a cocompact discrete subgroup. We construct fiber bundles over arithmetic varieties whose fibers are isomorphic to a twisted torus, and express the complex cohomology of such bundles over certain Riemann surfaces in terms of automorphic forms.

We show that a cocompact hyperbolic group in dimension 3 is characterized by certain properties of its word metric which depend only on the group structure and not on any action on hyperbolic space. We prove a similar theorem for finite-volume hyperbolic groups in dimension 3.

In this article we define and study a notion of asymptotic rank for metric spaces and show in our main theorem that for a large class of spaces, the asymptotic rank is characterized by the growth of the higher filling functions. For a proper, cocompact, simplyconnected geodesic metric space of non-curvature in the sense of Alexandrov the asymptotic rank equals its Euclidean rank.

We exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation a fundamental domain and linear algebra. As applications, we compute Shimura curve parametrizations of elliptic curves over a totally real field, including the image of CM points, and equations for Shimura curves.