Finiteness and the falsification by fellow traveler property

We prove that groups enjoying the falsification by fellow traveler property are of type $F_3$, and have at most an exponential second order isoperimetric function.Comment: Appeared in 2002. I am putting all my past publications on arxi

CAT(0) is an algorithmic property

In this article we give an explicit algorithm which will determine, in a discrete and computable way, whether a finite piecewise Euclidean complex is non-positively curved. In particular, given such a complex we show how to define a boolean combination of polynomial equations and inequalities in real variables, i.e. a real semi-algebraic set, which is empty if and only if the complex is non-positively curved. Once this equivalence has been shown, the main result follows from a standard theorem in real algebraic geometry.Comment: (23 pages) To appear in Geometriae Dedicat

Some geodesic problems in groups

We consider several algorithmic problems concerning geodesics in finitely generated groups. We show that the three geodesic problems considered by Miasnikov et al [arXiv:0807.1032] are polynomial-time reducible to each other. We study two new geodesic problems which arise in a previous paper of the authors and Fusy [arXiv:0902.0202] .Comment: 6 page

Logspace computations for Garside groups of spindle type

M. Picantin introduced the notion of Garside groups of spindle type, generalizing the 3-strand braid group. We show that, for linear Garside groups of spindle type, a normal form and a solution to the conjugacy problem are logspace computable. For linear Garside groups of spindle type with homogenous presentation we compute a geodesic normal form in logspace.Comment: 22 pages; short version as v1. Terminolgy and title changed. In particular, in previous versions we called Garside groups of spindle type "rigid Garside groups

Combinatorial conditions that imply word-hyperbolicity for 3-manifolds

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 and the universal cover contains no isometrically embedded flat planes. The proof involves a mixture of computer computation and techniques from small cancellation theory.Comment: (21 pages) To appear in Topolog

A non-Hopfian almost convex group

In this article we prove that an "isometric multiple HNN-extension" of a group satisfying the falsification by fellow traveler property is almost convex. As a corollary, Wise's example of a CAT(0) non-Hopfian group is Almost convex.Comment: Appeared in 2004. I am putting all my past papers on arxi