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The aim of this text is to provide a clear description of the theory of Infra-nilmanifolds and their fundamental groups, the almost-Bieberbach groups. For most of the proofs of the results, we refer to the literature. Nevertheless, at…

This is a survey on upper and lower bounds for finite group actions on bounded surfaces, 3-dimensional handlebodies and closed handles, handlebodies in arbitrary dimensions and finite graphs (the common feature of these objects is that all…

To study a noncompact Riemannian manifold, it is often useful to find a compactification. We discuss several common compactifications and survey some recent results.

This is a survey of using Minsky machines to study algorithmic problems in semigroups, groups and other algebraic systems.

We prove that, given any smooth action of a compact quantum group (in the sense of \cite{rigidity}) on a compact smooth manifold satisfying some more natural conditions, one can get a Riemannian structure on the manifold for which the…

The purpose of this paper is to study the transitive group-groupoids.

We consider a purely algebraic result. Then given a circle or cyclic group of prime order action on a manifold, we will use it to estimate the lower bound of the number of fixed points. We also give an obstruction to the existence of…

The purpose of this paper is to survey the structure of closed and transitive transformation groups acting on a closed surface. In particular, we prove a number of relations between groups acting on the sphere that contain the rotation…

Let M be a closed connected oriented manifold admitting a non-zero degree map to a nilmanifold. In the first part of the paper we study effective finite group actions on M. In particular, we prove that Homeo(M) is Jordan, we bound the…

Definition of a smooth action of a CQG on a compact, smooth manifold is given and studied. It is shown that a smooth action is always injective. Furthermore A necessary and sufficient condition for a lift of the smooth action as a bimodule…

In May 2015, a conference entitled "Groups, Geometry, and 3-manifolds" was held at the University of California, Berkeley. The organizers asked participants to suggest problems and open questions, related in some way to the subject of the…

Beauville surfaces are a class of complex surfaces defined by letting a finite group $G$ act on a product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the…

We generalize the notion of isoperimetric profiles of finitely generated groups to their actions by measuring the boundary of finite subgraphings of the orbit graphing. We prove that like the classical isoperimetric profiles for groups,…

Manifolds occur naturally as configuration spaces of robotic systems. They provide global descriptions of local coordinate systems that are common tools in expressing positions of robots. The purpose of this survey is threefold. Firstly, we…

Fix a module M over a local ring R and a group action G on M, not necessarily R-linear. To understand how large is the G-orbit of an element z\in M one looks for the large submodules of M lying in Gz. We provide the corresponding…

We investigate continuous transitive actions of semitopological groups on spaces, as well as separately continuous transitive actions of topological groups.

The main result of the paper is the complete classification of the compact connected Lie groups acting coisotropically on complex Grassmannians. This is used to determine the polar actions on the same manifolds.

: Algebraic properties of orbifold models on arbitrary Riemann surfaces are investigated. The action of mapping class group transformations and of standard geometric operations is given explicitly. An infinite dimensional extension of the…

In the present paper, we study several complex manifolds by using the following idea. First, we construct a certain moduli space and study the fundamental group of this space. This fundamental group is naturally mapped to the groups…

A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded…