arXiv subjects

Peter Brinkmann

Publications and source records attributed to Peter Brinkmann.

9 recordsLinked to original sources

arXiv2012

Polynomial invariants of pseudo-Anosov maps

We investigate the structure of the characteristic polynomial det(xI-T) of a transition matrix T that is associated to a train track representative of a pseudo-Anosov map [F] acting on a surface. As a result we obtain three new polynomial invariants of [F], one of them being the product of the other two, and all three being divisors of det(xI-T). The degrees of the new polynomials are invariants of [F ] and we give simple formulas for computing them by a counting argument from an invariant train track. We give examples of genus 2 pseudo-Anosov maps having the same dilatation, and use our invariants to distinguish them.

math.GT↗

arXiv2008

Dynamics of free group automorphisms

We present a coarse convexity result for the dynamics of free group automorphisms: Given an automorphism $ϕ$ of a finitely generated free group $F$, we show that for all $x\in F$ and $0\leq i\leq N$, the length of $ϕ^i(x)$ is bounded above by a constant multiple of the sum of the lengths of $x$ and $ϕ^N(x)$, with the constant depending only on $ϕ$.

math.GR↗

arXiv2008

Detecting automorphic orbits in free groups

We present an effective algorithm for detecting automorphic orbits in free groups, as well as a number of algorithmic improvements of train tracks for free group automorphisms.

math.GR↗

arXiv2003

A note on pseudo-Anosov maps with small growth rate

We present an explicit sequence of pseudo-Anosov maps $ϕ_k: S_{2k}\to S_{2k}$ of surfaces of genus $2k$ whose growth rates converge to one.

math.GT↗

arXiv2002

Splittings of mapping tori of free group automorphisms

We present necessary and sufficient conditions for the existence of a splitting over $\mathbb{Z}$ of the mapping torus $M_ϕ=F\rtimes \mathbb{Z}$ of a free group automorphism $ϕ$.

math.GR↗

arXiv2001

Computing Triangulations of Mapping Tori of Surface Homeomorphisms

We present the mathematical background of a software package that computes triangulations of mapping tori of surface homeomorphisms, suitable for Jeff Weeks's program SnapPea. It consists of two programs. jmt computes triangulations and prints them in a human-readable format. jsnap converts this format into SnapPea's triangulation file format and may be of independent interest because it allows for quick and easy generation of input for SnapPea. As an application, we obtain a new solution to the restricted conjugacy problem in the mapping class group.

math.GT↗

arXiv1999

Hyperbolic automorphisms of free groups

We prove that an automorphism $ϕ:F\to F$ of a finitely generated free group $F$ is hyperbolic in the sense of Gromov if it has no nontrivial periodic conjugacy classes.

math.GR↗

arXiv1999

Perimeter and Coherence According to McCammond and Wise

McCammond and Wise have introduced the perimeter of 2-complexes, and they have used it to obtain a sufficient criterion for coherence. We present a new exposition of their main result, as well as some applications.

math.GR↗

arXiv1999

An Implementation of the Bestvina-Handel Algorithm for Surface Homeomorphisms

Bestvina and Handel have found an effective algorithm that determines whether a given homeomorphism of an orientable, possibly punctured surface is pseudo-Anosov. We present a software package in Java that realizes this algorithm for surfaces with one puncture. Moreover, the package allows the user to define homeomorphisms in terms of Dehn twists, and in the pseudo-Anosov case it generates images of train tracks in the sense of Bestvina-Handel.

math.GR↗