Expanding endomorphisms of crystallographic manifolds

AbstractLet Γ be a crystallographic group with associated exact sequence 0→A→Γ→G→1, and let MΓ be the flat crystallographic manifold (i.e., the G-equivariant torus RnA associated to Γ. We construct a new crystallographic group Δ, a quotient of ΓAc, where Ac is the sum of all 1-dimensional G-submodules of A. Then we generalize the results of D. Epstein and M. Shub (1968) by showing the existence of equivariant endomorphisms of MΓ which expand distances in certain directions transverse to the fibers of the map MΓ→MΔ. The existence of such expanding maps is of interest to the study of the K-theory as well as the controlled K-theory of Γ

On the Mapping class group of a genus 2 handlebody

A complex of incompressible surfaces in a handlebody is constructed so that it contains, as a subcomplex, the complex of curves of the boundary of the handlebody. For genus 2 handlebodies, the group of automorphisms of this complex is used to characterize the mapping class group of the handlebody. In particular, it is shown that all automorphisms of the complex of incompressible surfaces are geometric, that is, induced by a homeomorphism of the handlebody

Geometries on Polygons in the unit disc

For a family $\mathcal{C}$ of properly embedded curves in the 2-dimensional disk $\mathbb{D}^{2}$ satisfying certain uniqueness properties, we consider convex polygons $P\subset \mathbb{D}^{2}$ and define a metric $d$ on $P$ such that $(P,d)$ is a geodesically complete metric space whose geodesics are precisely the curves $\left\{ c\cap P\bigm\vert c\in \mathcal{C}\right\}.$ Moreover, in the special case $\mathcal{C}$ consists of all Euclidean lines, it is shown that $P$ with this new metric is not isometric to any convex domain in $\mathbb{R} ^{2}$ equipped with its Hilbert metric. We generalize this construction to certain classes of uniquely geodesic metric spaces homeomorphic to $\mathbb{R}^{2}.$Comment: To appear in Rocky Mountain J. Mat

A specific model of Hilbert geometry on the unit disc

A new metric on the open 2-dimensional unit disk is defined making it a geodesically complete metric space whose geodesic lines are precisely the Euclidean straight lines. Moreover, it is shown that the unit disk with this new metric is not isometric to any hyperbolic model of constant negative curvature, nor to any convex domain in R2 equipped with its Hilbert metric.Comment: To appear in Beitr Algebra Geo