### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 134, 18 pages      arXiv:1804.09603      https://doi.org/10.3842/SIGMA.2018.134

### A Product on Double Cosets of $B_\infty$

Pablo Gonzalez Pagotto
Institut Fourier, Université Grenoble Alpes, Grenoble, France

Received May 28, 2018, in final form December 14, 2018; Published online December 27, 2018

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.

Key words: Braid group; double cosets; Burau representation.

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