arXiv subjects

Joseph Maher

Publications and source records attributed to Joseph Maher.

25 records · Page 2Linked to original sources

arXiv2010

Random Heegaard splittings

A random Heegaard splitting is a 3-manifold obtained by using a random walk of length n on the mapping class group as the gluing map between two handlebodies. We show that the joint distribution of random walks of length n and their inverses is asymptotically independent, and converges to the product of the harmonic and reflected harmonic measures defined by the random walk. We use this to show that the translation length on the curve complex of a random walk grows linearly in the length of the walk, and similarly, that distance in the curve complex between the disc sets of a random Heegaard splitting grows linearly in n. In particular, this implies that a random Heegaard splitting is hyperbolic with asymptotic probability one.

math.GT↗

arXiv2010

Asymptotics for pseudo-Anosov elements in Teichmuller lattices

A Teichmuller lattice is the orbit of a point in Teichmuller space under the action of the mapping class group. We show that the proportion of lattice points in a ball of radius r which are not pseudo-Anosov tends to zero as r tends to infinity. In fact, we show that if R is a subset of the mapping class group, whose elements have an upper bound on their translation length on the complex of curves, then proportion of lattice points in the ball of radius r which lie in R tends to zero as r tends to infinity.

math.GT↗

arXiv2010

Linear progress in the complex of curves

We show that a random walk on the mapping class group of an orientable surface of finite type makes linear progress in the relative metric, which is quasi-isometric to the complex of curves.

math.GT↗

arXiv2008

Period three actions on lens spaces

We show that a free period three action on a lens space is standard, i.e. the quotient is homeomorphic to a lens space. This is an extension of the result for period three actions on the three-sphere, arXiv:math.GT/0204077, by the author and J. Hyam Rubinstein.

math.GT↗

arXiv2005

Heegaard gradient and virtual fibers

We show that if a closed hyperbolic 3-manifold has infinitely many finite covers of bounded Heegaard genus, then it is virtually fibered. This generalizes a theorem of Lackenby, removing restrictions needed about the regularity of the covers. Furthermore, we can replace the assumption that the covers have bounded Heegaard genus with the weaker hypotheses that the Heegaard splittings for the covers have Heegaard gradient zero, and also bounded width, in the sense of Scharlemann-Thompson thin position for Heegaard splittings.

math.GT↗

arXiv2003

Period three actions on the three-sphere

We show that a free period three action on the three-sphere is standard, i.e. the quotient is homeomorphic to a lens space. We use a minimax argument involving sweepouts.

math.GT↗

arXiv1999

Virtually embedded boundary slopes

We show that for certain hyperbolic 3-manifolds, all boundary slopes are slopes of immersed incompressible surfaces, covered by incompressible embeddings in some finite cover. The manifolds include hyperbolic punctured torus bundles and hyperbolic two-bridge knots.

math.GT↗