Finite and infinite frieze patterns from p-angulations and a generalization of Weyl groupoids

A classic result of Conway and Coxeter on frieze patterns has been
generalized to a bijection between $p$-angulations of regular polygons and
frieze patterns of type $\Lambda_p$. One of the features of Conway-Coxeter
theory is a combinatorial procedure to obtain from the triangulation all
entries of the corresponding frieze pattern. We first present a combinatorial
algorithm, involving Chebyshev polynomials, for obtaining from a dissection all
entries of the corresponding frieze pattern. As an application we obtain a
characterisation of frieze patterns of types $\Lambda_4$ and $\Lambda_6$ in
terms of all entries (not only the quiddity cycle). We then study infinite
frieze patterns of type $\Lambda_p$, which appeared in a preprint by Banaian
and Chen, generalizing the infinite frieze patterns of positive integers
studied by Baur, Parsons and Tschabold. As our main result we obtain a
combinatorial model for infinite frieze patterns of type $\Lambda_p$, these are
in bijection with certain $p$-angulations of an infinite strip. This extends
results by Baur, Parsons and Tschabold from $p=3$ to arbitrary $p\ge 3$, and
also provides new insight in the classic case. Infinite frieze patterns of
positive integers appear in the context of Weyl groupoids. In the final section
we extend this to infinite frieze patterns of type $\Lambda_p$ for any $p\ge 3$
by introducing a generalization of Cartan graphs and Weyl groupoids. We show
that, up to equivalence, there is a 1-1 correspondence between connected simply
connected Cartan graphs of type $\Lambda_p$ of rank two with infinitely many
vertices permitting a root system and infinite frieze patterns of type
$\Lambda_p$.