On a Garside group extending the braid group

23 pages, 3 figures, 27 ref.; The submonoid of the $3$-strand braid group $\mathcal{B}_3$ generated by $\sigma_1$ and $\sigma_1 \sigma_2$ is known to yield an exotic Garside structure on $\mathcal{B}_3$. We introduce and study an infinite family $(M_n)_{n\geq 1}$ of Garside monoids generalizing this exotic Garside structure, i.e., such that $M_2$ is isomorphic to the above monoid. The corresponding Garside group $\mathcal{G}_n$ is an extension of the $(n+1)$-strand braid group, isomorphic to $\mathcal{B}_3$ when $n=2$ and to the complex braid group of the complex reflection group $G_{12}$ when $n=3$. In general, the Garside monoid $M_n$ sujects onto the submonoid $\Sigma_n$ of $\mathcal{B}_{n+1}$ generated by $\sigma_1, \sigma_1 \sigma_2, \dots, \sigma_1 \sigma_2\cdots \sigma_n$, which is not a Garside monoid when $n>2$. Using a new presentation of $\mathcal{B}_{n+1}$ similar to the presentation of $\mathcal{G}_n$, we nevertheless check that $\Sigma_n$ is an Ore monoid with group of fractions isomorphic to $\mathcal{B}_{n+1}$, and give a conjectural presentation of it. We also show that the groups $\mathcal{G}_n$ are isomorphic to groups defined by a cyclic relation; such groups were shown to be Garside groups by Dehornoy and Paris. As a byproduct the monoid $M_n$ yields a new Garside structure for these groups.