Uniform difference between the length spectra of Out(F2) and the genus two handlebody group

TL;DR

This paper investigates the length spectra relationship between the genus two handlebody group and outer automorphisms of free groups, revealing bounds on stretch factors and implications for geodesic counting.

Contribution

It establishes a uniform bound relating stretch factors of pseudo-Anosov maps and fully irreducible outer automorphisms in genus two, addressing a question by Hensel.

Findings

Minimum stretch factor in genus two is less than ten times that of the outer automorphism.

A lower bound for the geodesic counting problem in the genus two handlebody group.

Partial resolution of Hensel's question on length spectra relations.

Abstract

In this paper, we analyze the natural homomorphism from the genus g handlebody group to the outer automorphism group of the free group with rank g, in terms of length spectra. In general, the preimage of each fully irreducible outer automorphism contains a potentially infinite number of pseudo-Anosov mapping classes. Our study reveals a crucial relationship: for any pseudo-Anosov map in this preimage, its stretch factor must equal or exceed that of the corresponding fully irreducible outer automorphism. Notably, in the case of genus two, we establish that the minimum stretch factor among these pseudo-Anosov maps is less than ten times the stretch factor of the fully irreducible outer automorphism. These results partially address a question by Hensel and have practical implications, including a lower bound for the geodesic counting problem in the genus two handlebody group.