arXiv subjects

Murray Elder

Publications and source records attributed to Murray Elder.

61 records · Page 4Linked to original sources

arXiv2005

Cone types and geodesic languages for lamplighter groups and Thompson's group F

We study languages of geodesics in lamplighter groups and Thompson's group F. We show that the lamplighter groups $L_n$ have infinitely many cone types, have no regular geodesic languages, and have 1-counter, context-free and counter geodesic languages with respect to certain generating sets. We show that the full language of geodesics with respect to one generating set for the lamplighter group is not counter but is context-free, while with respect to another generating set the full language of geodesics is counter and context-free. In Thompson's group F with respect to the standard finite generating set, we show there are infinitely many cone types and no regular language of geodesics with respect to the standard finite generating set. We show that the existence of families of "seesaw" elements with respect to a given generating set in a finitely generated infinite group precludes a regular language of geodesics and guarantees infinitely many cone types with respect to that generating set.

math.GR↗

arXiv2005

Pattern avoiding permutations are context-sensitive

We establish a bijection from the set of all permutations (of a given length) that avoid a pattern $q$ and a context-sensitive language.

math.CO↗

arXiv2005

Regular geodesic languages and the falsification by fellow traveler property

We furnish an example of a finite generating set for a group that does not enjoy the falsification by fellow traveler property, while the full language of geodesics is regular.

math.GR↗

arXiv2004

A context-free and a 1-counter geodesic language for a Baumslag-Solitar group

We give a language of unique geodesic normal forms for the Baumslag-Solitar group BS(1,2) that is context-free and 1-counter. We discuss the classes of context-free, 1-counter and counter languages, and explain how they are inter-related.

math.GR↗

arXiv2004

L_delta groups are almost convex and have a sub-cubic Dehn function

We prove that if the Cayley graph of a finitely generated group enjoys the property L_delta then the group is almost convex and has a sub-cubic isoperimetric function.

math.GR↗

arXiv2003

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.

math.GT↗

arXiv2003

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.

math.GT↗