group associated to the complex reflection group $G(e,e,r)$. The ${{\mathrm{B}}}(e,e,r)$ are the group of fractions of a Garside monoid introduced by R. Corran and M. Picantin (see [@corpic]). This monoid, that we denote $M(e,e,r)$, has generators (atoms) $t_0,t_1,\dots,t_{e-1}, s_3, s_4, \dots, s_r$ and relations

1.  $t_{i+1} t_i = t_{j+1} t_j$, with the convention $t_e = t_0$,

2.  $s_3 t_i s_3 = t_i s_3 t_i$

3.  $s_k t_i  = t_i s_k$ for $k \geq 4$

4.  $s_k s_{k+1} s_k = s_{k+1} s_k s_{k+1} $ for $k \geq 3$

5.  $s_k s_l = s_l s_k$ when $| l-k | \geq 2$.

Link with the topological definition
------------------------------------

The connection between this monoid and the group ${{\mathrm{B}}}(e,e,r)$ defined as a fundamental group is quite indirect. In [@BMR] a first presentation is obtained by combining embeddings into usual braid groups, fibrations and coverings. The presentation used here is deduced from this one in a purely algebraic matter, by adding generators in order to get a Garside presentation. Although it is folklore the description of all generators as braided reflection does not appear in the literature (see however [@BESSISCORRAN] for a statement without proof in a related context).

In order to provide this connection, we 