## Note on a theorem of Munkres.(English)Zbl 1227.57032

Summary: We prove that given a $${\mathcal C}^\infty$$ Riemannian manifold with boundary, having a finite number of compact boundary components, any fat triangulation of the boundary can be extended to the whole manifold. We also show that this result extends to $${\mathcal C}^1$$ manifolds and to embedded $$PL$$ manifolds of dimensions 2, 3 and 4. We employ these results to prove that manifolds of the types above admit quasimeromorphic mappings onto $$\widehat{{\mathbb R}^n}$$. As an application, we prove the existence of $$G$$-automorphic quasimeromorphic mappings, where $$G$$ is a Kleinian group acting on $$\mathbb H^n$$.

### MSC:

 57R05 Triangulating 57M60 Group actions on manifolds and cell complexes in low dimensions 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations

### Keywords:

fat triangulation; quasimeromorphic mapping
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