arXiv research

New hyperbolic 3-pseudomanifolds with unique properties.

Research explores hyperbolic space groups and their fundamental domains.

New proof shows amalgamations of hyperbolic 3-manifold groups are not LERF.

Proves hyperbolized groups are virtually compact special and linear.

Finite height in relatively hyperbolic groups ensures relative quasiconvexity.

The paper defines fundamental domains for Fuchsian groups in hyperbolic space.

Generalizes Nielsen equivalence theorem to hyperbolic group extensions.

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of $2π$, determines a holonomy representation …

Study complex hyperbolic lattices and their relation to strict hyperbolization.

New representations of 3-manifold groups into complex hyperbolic space found.

Paper shows 3D hyperbolic manifolds are uniquely identified by their finite groups.

New examples of hyperbolic 3-manifolds with unique profinite structure.

Anabelian geometry reformulated using Hodge theory for hyperbolic curves.

The study proves stabilizing of ascending chains in specific groups.

We show that the aspherical manifolds produced via the relative strict hyperbolization of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automor…

Study shows how hyperbolic groups relate to Sierpinski spaces.

New examples of 5D manifolds without certain structures.

A 5-manifold fibers over a circle with nonpositive curvature.

Generators found for a complex hyperbolic braid group.

Proves strong version of Hopf problem for hyperbolic circle bundles.

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of $2π$, determines a holonomy representation …

Proves 3-manifold groups uniquely identify hyperbolic bundles.

We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n-1)-submanifold. The main result is that the fundamental group of M-S is relatively hyperbolic, relative to fundamental groups of the ends of M-S, and M-S admits a complete finite volu…

The paper shows how to find inaccessible hyperbolic actions in manifold groups.

Groups with semistable peripheral subgroups are semistable.

The paper proves non-LERFness for certain manifold groups and explores abelian amalgamations.

We show that there is an upper bound on the injectivity radius of a hyperbolic 3-manifold in terms of the the number of generators of its fundamental group.

The paper studies a group action on a hyperbolic space derived from a lattice Veech group.

By using Thurston's bending construction we obtain a sequence of faithful discrete representations ρ_n of the fundamental group of a closed hyperbolic 3-manifold fibering over the circle into the isometry group Iso H^4 of the hyperbolic space H^4. The algebraic limit of ρ_n contains a finitely generated subgroup F whos…

In this paper, we study the fundamental group of a certain class of globally hyperbolic Lorentzian manifolds with a positive curvature tensor. We prove that the fundamental group of lightlike geodesically complete parametrized Lorentzian products is finite under the conditions of a positive curvature tensor and the fib…

A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are: * If a symplectic form represents a bounded cohomology cl…

New hyperbolic groups found that are not coHopfian.

Develop criteria to distinguish Gromov-Thurston manifolds using algebraic Dehn fillings.

Suppose n>2, let M,M' be n-dimensional connected complete finite-volume hyperbolic manifolds with non-empty geodesic boundary, and suppose that the fundamental group of M is quasi-isometric to the fundamental group of M' (with respect to the word metric). Also suppose that if n=3, then the boundaries of M and of M' are…

New groups with special properties found.

We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional hyperbolic polyhedral orbifold to contain an immersed (singular) hyperbolic tur…

Lower bounds for quaternionic hyperbolic orbifold volumes found.

Explains hyperbolic group boundaries using amalgams.

Study on fundamental 3-classes of knot representations using diagrams and algorithms.

The paper uses Tannakian reconstruction to understand hyperbolic log-orbi curves.

Characterizes fundamental groups of disjointly tree-graded spaces.

The study finds that certain hyperbolic manifolds contain subgroups isomorphic to surface groups.

Researchers found only one hyperbolic structure for Borromean rings.

This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M^n with hyperbolic fundamental group is a retract of …

Proves regularity of isomorphisms between hyperbolic 3-manifolds.

Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [Hyperbolic groups, MSRI publications 8 (1987)]. Using the definition of Farb of a relatively hyperbolic group in the strong sense [B Farb, Relatively hyperbolic groups, Geom. Func. Anal. 8 (1998) 810-840], we prove this asser…

Let O be a compact orientable 3-orbifold with non-empty singular locus and a finite volume hyperbolic structure. (Equivalently, O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {O_i} with linear growth of p-homology, for some prime…

New method detects left-orderable surgeries on knot 6_2.