Property $P_{\text{naive}}$ for big mapping class groups

TL;DR

This paper investigates a specific algebraic property, $P_{naive}$, in the context of infinite-type surface mapping class groups, exploring how certain elements can generate free products with given elements.

Contribution

It introduces and analyzes the property $P_{naive}$ for big mapping class groups, providing new insights into their algebraic structure.

Findings

Established conditions under which $P_{naive}$ holds

Identified classes of infinite-type surfaces with property $P_{naive}$

Contributed to understanding the algebraic complexity of big mapping class groups

Abstract

We study the property $P_{\text {naive }}$ of mapping class groups of surfaces of infinite type, that is, for any finite collection of non-trivial elements $h_{1},h_{2}, \cdots, h_{n}$, there exists another element $g\neq 1$ of infinite order such that for all $i$, $\langle g, h_{i}\rangle \cong \langle g \rangle * \langle h_{i} \rangle$.