On dilatation factors of braids on three strands

TL;DR

This paper introduces a surjective map from rigid 3-strand braids to SL_2(N), providing bounds on dilatation factors of pseudo-Anosov braids based on Garside structure length.

Contribution

It establishes a novel connection between rigid braids and SL_2(N), offering bounds on dilatation factors that depend solely on Garside canonical length.

Findings

Bounds on dilatation factors derived from Garside length

Surjective map from rigid braids to SL_2(N)

Applicable to pseudo-Anosov 3-strand braids

Abstract

In this work we present a natural surjective map from rigid braids in B_3 (in Garside sense) to SL_2(N). This map provides an upper and a lower bound for the dilatation factor of a pseudo-Anosov 3-strand braid. These bounds only depend on the canonical length of the classical Garside structure of B_3.