A Product on Double Cosets of $B_\infty$

TL;DR

This paper demonstrates that the infinite braid group $B_ty$ can be endowed with a monoid structure on its double cosets, expanding understanding of algebraic structures in infinite-dimensional groups.

Contribution

It establishes a monoid structure on the double cosets of the infinite braid group $B_ty$, a novel result in the study of infinite-dimensional groups.

Findings

$B_ty$ admits a monoid structure on its double cosets.

The structure extends the algebraic framework of finite braid groups.

Provides new tools for studying infinite-dimensional group symmetries.

Abstract

For some infinite-dimensional groups $G$ and suitable subgroups $K$ there exists a monoid structure on the set $K\backslash G/K$ of double cosets of $G$ with respect to $K$. In this paper we show that the group $B_\infty$, of the braids with finitely many crossings on infinitely many strands, admits such a structure.