arXiv research

Word embeddings in hyperbolic space outperform Euclidean ones.

We describe a procedure which verifies that a group given by generators and relators is word-hyperbolic. This procedure always works with a group which is word-hyperbolic, provided there is sufficient memory and time devoted to the problem. If the group is not word-hyperbolic, the procedure continues indefinitely. We a…

A longstanding question of Gromov asks whether every one-ended word-hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. An infinite family of word-hyperbolic groups can be obtained by taking doubles of free groups amalgamated along words that are not proper powers. W…

We simplify word embeddings by removing sigmoid in SGNS, revealing connections to hyperbolic spaces.

Paper proves $C^0$-semi-rigidity of meandering-hyperbolic actions.

Groups satisfy linear surface isoperimetric functions.

We examine residual properties of word-hyperbolic groups, adapting a method introduced by Darren Long to study the residual properties of Kleinian groups.

Study projection in acylindrically hyperbolic groups, proving sublinear tracking and growth bounds.

Proves restrictions on projective Anosov representations of hyperbolic groups.

We prove an equivariant version of the fact that word-hyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the Farrell-Jones conjecture in algebraic K-theory for every word-hyperbolic group G and every coefficient ring R.

Classifies complex hyperbolic triangle groups by types.

New definitions and properties for hyperbolic group representations.

New spectral Dehn function characterizes word-hyperbolic groups.

A new model embeds word and label hierarchies in hyperbolic space for HMLC.

We outline a rigorous algorithm, first suggested by Casson, for determining whether a closed orientable 3-manifold M is hyperbolic, and to compute the hyperbolic structure, if one exists. The algorithm requires that a procedure has been given to solve the word problem in π_1(M).

Proves involutions on Right-angled Coxeter groups without fixed points.

We define families of aperiodic words associated to Lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots. An algorithm to construct symbolic words of satellite Lorenz knots is defined. We prove, subject to the validity of a previous conjecture, that Lorenz knots code…

Each free homotopy class of directed closed curves on a surface with boundary can be described by a cyclic reduced word in the generators of the fundamental group and their inverses. The word length is the number of letters of the cyclic word. If the surface has a hyperbolic metric with geodesic boundary, the geometric…

The paper develops algorithms to detect stability and Morse properties in various groups.

Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special cases, including W word hyperbolic, and X with isolated flats. We establish tha…

Study proves hyperbolic groups have specific subgroup properties.

There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichm{ü}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of su…

New findings on algebraic structure of hyperbolic graph braid groups.

We show that for a hyperbolic knot complement, all but at most 12 Dehn fillings are irreducible with infinite word-hyperbolic fundamental group.

Paper introduces a hyperbolic Gaussian distribution for better learning in hierarchical data.

The aim of this paper is to demonstrate that very many Dehn fillings on a cusped hyperbolic 3-manifold yield a 3-manifold which is irreducible, atoroidal and not Seifert fibred, and which has infinite, word hyperbolic fundamental group. We establish an extension of the Thurston-Gromov $2π$ theorem by showing that if ea…

Let S be a closed surface of genus at least 2. We show that a finitely generated group G which is an extension of the fundamental group H of S is word hyperbolic if and only the orbit map of the quotient group G/H on the complex of curves is a quasi-isometric embedding.This in turn is equivalent to G/H being convex coc…

The paper characterizes convex co-compact groups with one-dimensional boundary faces.

Study convex cocompact subgroups in real projective geometry.

It is shown that a hyperbolic knot in the 3-sphere admits at most nine integral surgeries yielding 3-manifolds which are reducible or whose fundamental groups are not infinite word-hyperbolic.

We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.

Thurston conjectured that a closed triangulated 3-manifold in which every edge has degree 5 or 6, and no two edges of degree 5 lie in a common 2-cell, has word-hyperbolic fundamental group. We establish Thurston's conjecture by proving that such a manifold admits a piecewise Euclidean metric of non-positive curvature a…

The Manhattan curve connects metrics of hyperbolic groups, showing rigidity.

The paper constructs hyperbolic elements in multiple spaces.

New examples show embeddings not approximated by Anosov representations.

Algorithm decides if geodesic curves are filling on surfaces.

Exponential proportion of pseudo-Anosovs in mapping class groups.

We prove a Morse Lemma for coarsely regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings X. The main application is a simpler coarse geometric characterization of Morse subgroups of the isometry groups Isom(X) as undistorted subgroups which are coarsely uniformly regular. We show furt…

Algorithm solves word problem for 3-manifold groups.

We investigate the average-case complexity of decision problems for finitely generated groups, in particular the word and membership problems. Using our recent results on ``generic-case complexity'' we show that if a finitely generated group $G$ has the word problem solvable in subexponential time and has a subgroup of…

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups $Q_1$ and $Q_2$ is relatively quasiconvex and isomorphic to $Q_1 \ast_{Q_1 \cap Q_2} Q_2$. The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, an…

Paper proposes a novel method for aligning hierarchical data using optimal transport in hyperbolic spaces.

Let $G$ be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space $Z$ so that there exists a continuous $G$-equivariant map $i:\partial G\to Z$, which we call a \emph{Cannon-Thurston map}. We obtain two characterzations (a dynamical one and a geometric one) of conical limit p…

Study geodesics in hyperbolic surfaces and trees, proving edge length properties.

Abstract Coxeter groups have growth rates that are Perron numbers.

The study shows that certain groups can be uniquely identified by their finite abelian summands.

Generic loxodromic elements are common in hyperbolic groups and grow linearly.

Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on H_2(G;R) is a finite-sided rational polyhedron.