Shifts maps are not type-preserving

TL;DR

The paper demonstrates that combining Dehn twists with shift maps on certain surfaces produces loxodromic isometries, proving that shift maps do not preserve the type of isometries in the relative arc graph.

Contribution

It shows that shift maps, when composed with Dehn twists, generate loxodromic isometries, revealing that shift maps are not type-preserving on the relative arc graph.

Findings

Composing Dehn twists with shift maps yields loxodromic isometries.

Shift maps are not type-preserving in the context of the relative arc graph.

The result applies to surfaces with an isolated puncture admitting a shift map.

Abstract

For a surface $S$ of sufficient complexity, Dehn twists act elliptically on the arc, curve, and relative arc graph of $S$. We show that composing a Dehn twist with a shift map results in a loxodromic isometry of the relative arc graph $\mathcal{A}(S,p)$ for any surface $S$ with an isolated puncture $p$ admitting a shift map. Therefore, shift maps are not type-preserving.