Crystallographic groups arising from Teichmüller spaces (Q2328028)

The article under review shows that certain crystallographic groups appear at maximally degenerated frontier points of the augmented Teichmüller spaces. Let $\Sigma_{g,n}$ be a closed oriented topological surface of genus $g$ with $n$ points deleted. Moreover, let $T_{g,n}$ denote the Teichmüller space of $\Sigma_{g,n}$ and let $\overline{T}_{g,n}$ be the Weil-Petersson completion of $T_{g,n}$. Let $\mathrm{Aut}(T_{g,n})$ denote the group of self-biholomorphisms of $T_{g,n}$. A natural homomorphism $\alpha : \Gamma_{g,n}\to\mathrm{Aut}(T_{g,n})$ is defined by the action of $\Gamma_{g,n}$ on $T_{g,n},$ here $\Gamma_{g,n}$ is the mapping class group. Thus the group $\mathrm{Aut}(T_{g,n})$ acts on the completion $\overline{T}_{g,n}$. Let $\mathcal D$ be a pants decomposition of $\Sigma_{g,n}$. Then we have a unique frontier point $p(\mathcal D)$ corresponding to the degenerate surface obtained by pinching each curve in $\mathcal D$ to a node. The following two theorems are the main results of the paper under review. 1. Suppose $m = 3g-3+n > 1$. Then for the action of $\mathrm{Aut}(T_{g,n})$ on $\overline{T}_{g,n}$, the isotropy group of the frontier point $p(\mathcal D)$ is a crystallographic group acting on Euclidean $m$-space $E^{m}$. 2. Let $m = 3g-3+n > 1, (g,n)\neq (2,0),(1,2)$. Then to each pants decomposition $\mathcal D$ of $\Sigma_{g,n}$ a subgroup of $\Gamma_{g,n}$ is associated which is a crystallographic group acting on Euclidean $m$-space $E^{m}$. In the first part of this interesting paper, definitions and properties of Teichmüller spaces and crystallographic groups are reviewed. The main results are proved in the fifth section. Some examples are presented in the last part 6 of the article.