Related papers: Problems on Bieberbach groups and flat manifolds

This is a summary of some of the basic facts about flat 2-orbifold groups, otherwise known as 2-dimensional crystallographic groups. We relate the geometric and topological presentations of these groups, and consider structures…

Deformation theory of complex manifolds is a classical subject with recent new advances in the noncompact case using both algebraic and analytic methods. In this note, we recall some concepts of the existing theory and introduce new notions…

We give a characterisation of Bieberbach manifolds which are geodesic boundaries of a compact flat manifold, and discuss the low dimensional cases, up to dimension 4.

Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject were collected in author's books [26,29]. In 1991, a list of twelve open problems and three conjectures on finite type…

We collect a few guesses on possible implications of a lower bound on the scalar curvature of a Riemannian manifold on the size and shape of this manifold.

We provide, for hyperbolic and flat 3-manifolds, obstructions to bounding hyperbolic 4-manifolds, thus resolving in the negative a question of Farrell and Zdravkovska.

Solutions to the Riemann-Hilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The…

This survey focuses on the computational complexity of some of the fundamental decision problems in 3-manifold theory. The article discusses the wide variety of tools that are used to tackle these problems, including normal and almost…

We show that there exist flat surface bundles with closed leaves having non-trivial normal bundles. This leads us to compute the Abelianisation of surface diffeomorphism groups with marked points. We also extend a formula of Tsuboi that…

This is a short survey on finite-volume hyperbolic four-manifolds. We describe some general theorems and focus on the concrete examples that we found in the literature. The paper contains no new result.

The theory of complex hyperbolic discrete groups is still in its childhood but promises to grow into a rich subfield of geometry. In this paper I will discuss some recent progress that has been made on complex hyperbolic deformations of the…

The class of the hypercomplex pseudo-Hermitian manifolds is considered. The flatness of the considered manifolds with the 3 parallel complex structures is proved. Conformal transformations of the metrics are introduced. The conformal…

The general construction of self-adjoint configuration space representations of the Heisenberg algebra over an arbitrary manifold is considered. All such inequivalent representations are parametrised in terms of the topology classes of flat…

A survey of problems, conjectures, and theorems about quasi-isometric classification and rigidity for finitely generated solvable groups.

It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algebras of abelian varieties of GL_2-type over \Q of bounded dimension. We explore this conjecture when particularized to quaternion endomorphism…

The present survey aims at being a list of Conjectures and Problems in an area of model-theoretic algebra wide open for research, not a list of known results. To keep the text compact, it focuses on structures of finite Morley rank,…

We realize 4 of the 6 closed orientable flat 3-manifolds as a cusp section of an orientable finite-volume hyperbolic 4-manifold whose symmetry group acts transitively on the set of cusps.

: 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…

Is the Universe (a spatial section thereof) finite or infinite? Knowing the global geometry of a Friedmann-Lema\^{\i}tre (FL) universe requires knowing both its curvature and its topology. A flat or hyperbolic (``open'') FL universe is {\em…

Some conjectures about Heegaard genera and ranks of fundamental groups of 3-manifolds are formulated, and it is shown that they imply new statements about hyperbolic volume.