相关论文: Problems on Bieberbach groups and flat manifolds

We discuss a number of open problems about mapping class groups of surfaces. In particular, we discuss problems related to linearity, congruence subgroups, cohomology, pseudo-Anosov stretch factors, Torelli subgroups, and normal subgroups.

We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with…

We remark some basic facts on homological aspects of involutive Lie bialgebras and their involutive bimodules, and present some problems on surface topology related to these facts.

Following an idea of Gon\c{c}alvez, Guaschi and Ocampo on the usual braid group we construct crystallographic and Bieberbach groups as (sub)quotients of the generalized braid group associated to an arbitrary complex reflection group.

The topic of this survey is the phenomenon of fibring over the circle for manifolds, and its group-theoretic twin, algebraic fibring. We will discuss the state of the art, and explain briefly some of the ideas behind the more recent…

Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a flat metric with several cone-type…

Carrier graphs of groups representing subgroups of a given relatively hyperbolic groups are introduced and a combination theorem for relatively quasi-convex subgroups is proven. Subsequently a theory of folds for such carrier graphs is…

We classify and construct all real spectral triples over noncommutative Bieberbach manifolds, which are restrictions of irreducible real equivariant spectral triple over the noncommutative three-torus. We show that in the classical case the…

We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true…

In this paper we study flat deformations of real subschemes of $\mathbb{P}^n$, hyperbolic with respect to a fixed linear subspace, i.e. admitting a finite surjective and real fibered linear projection. We show that the subset of the…

Geometric structures on manifolds became popular when Thurston used them in his work on the geometrization conjecture. They were studied by many people and they play an important role in higher Teichm\"uller theory. Geometric structures on…

This is a selection of open problems dealing with ``large'' (non locally compact) topological groups and concerning extreme amenability (fixed point on compacta property), oscillation stability, universal minimal flows and other aspects of…

A group is metabelian if its commutator subgroup is abelian. For finitely generated metabelian groups, classical commutative algebra, algebraic geometry and geometric group theory, especially the latter two subjects, can be brought to bear…

The study of algebraic properties of groups of transformations of a manifold gives rise to an interplay between different areas of mathemathics such as topology, geometry, and dynamical systems. Especially, in this paper, we point out some…

A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the…

Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject had been collected in author's book [Total mean curvature and sub manifolds of finite type, World Scientific, NJ, 1984]. A…

We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by…

We describe some open questions related to support points in the class $S^0$ and introduce some useful techniques toward a higher dimensional Bieberbach conjecture.

We present some constructions that are merely the fruit of combining recent results from two areas of smooth dynamics: nonuniformly hyperbolic systems and elliptic constructions.

We show the existence of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants, addressing a conjecture of Claude LeBrun. This is achieved by showing, using results in geometric and arithmetic group theory, that certain hyperbolic…