For positive integers $m$ and $k$, we introduce a family of lattices $\mathcal{ C}_{ k}^{ (m)}$ associated to the Cambrian lattice $\mathcal{ C}_{ k}$ of the dihedral group $I_{ 2}(k)$. We show that $\mathcal{ C}_{ k}^{ (m)}$ satisfies some basic properties of a Fuss-Catalan generalization of $\mathcal{ C}_{ k}$, namely that $\mathcal{ C}_{ k}^{ (1)}=\mathcal{ C}_{ k}$ and $\bigl\lvert\mathcal{ C}_{ k}^{ (m)}\bigr\rvert=\mbox{ Cat}^{ (m)}\bigl(I_{ 2}(k)\bigr)$. Subsequently, we prove some structural and topological properties of these lattices---namely that they are trim and EL-shellable---which were known for $\mathcal{ C}_{ k}$ before. Remarkably, our construction coincides in the case $k=3$ with the $m$-Tamari lattice of parameter 3 due to Bergeron and Pr{ \'e}ville-Ratelle. Eventually, we investigate this construction in the context of other Coxeter groups, in particular we conjecture that the lattice completion of the analogous construction for the symmetric group $\mathfrak{ S}_{ n}$ and the long cycle $(1\;2\;\ldots\;n)$ is isomorphic to the $m$-Tamari lattice of parameter $n$.
Categories: math.CO