Handlebody subgroups in a mapping class group

TL;DR

This paper investigates conditions under which subgroups of the mapping class group, associated with handlebodies, form free products with amalgamation, addressing a question posed by Minsky.

Contribution

It provides new insights into the structure of handlebody subgroups within the mapping class group and characterizes when their generated subgroup is an amalgamated free product.

Findings

Identifies conditions for handlebody subgroups to form free products with amalgamation.

Answers Minsky's question regarding subgroup structure in the mapping class group.

Provides criteria for subgroup isomorphism to amalgamated free products.

Abstract

Suppose subgroups $A,B < MCG(S)$ in the mapping class group of a closed orientable surface $S$ are given and let $\langle A, B \rangle$ be the subgroup they generate. We discuss a question by Minsky asking when $\langle A, B \rangle \simeq A*_{A \cap B} B$ for handlebody subgroups $A,B$.