In this paper, we develop the theory of “cuspidalizations” of the étale fundamental group of a proper hyperbolic curve over a finite or nonarchimedean mixed-characteristic local field. The ultimate goal of this theory is the group-theoretic reconstruction of the étale fundamental group of an arbitrary open subscheme of the curve from the étale fundamental group of the full proper curve. We then...

We show that if a closed irreducible three-manifold with hyperbolic fundamental group contains a surface subgroup satisfying a certain geometric regularity assumption, then the surface is either quasiconvex or a virtual fiber. In the latter case, the manifold is hyperbolic. The regularity condition ensures that we may find algebraic bounds on the surface group which are analogous to the diamete...

A complex hyperbolic triangle group is a group generated by three complex reflections fixing complex slices (complex geodesics) in complex hyperbolic space. Our purpose in this paper is to improve the result in [3] and to discuss discreteness of complex hyperbolic triangle groups of type (n, n,∞; k).

We study the countable set of rates growth a hyperbolic group with respect to all its finite generating sets. prove that is well-ordered, and every real number can be rate at most finitely many sets up automorphism group. ordinal least ${\omega _{0}}^{\omega _{0}}$ , in case limit (e.g. free surface groups) it . further generated subgroups their This proved well-ordered as well, isomorphism cla...