Dirichlet fundamental domains and topology of projective varieties
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We prove that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings and Whitney umbrellas.
KeywordsProjective Variety Fundamental Domain Normal Crossing Polyhedral Complex Cartan Involution
This paper grew out of our work  with János Kollár and I am grateful to him for questions, comments and suggestions. In particular, he explained to me that Whitney umbrella singularities appear as ℤ2-quotients of normal crossings and suggested the dimension reduction from 3 to 2. I am grateful to Akira Ushijima for sharing with me . I am also grateful to the referee for useful remarks and suggestions. Partial financial support for this work was provided by the NSF grants DMS-09-05802 and DMS-12-05312.
- 5.Bonahon, F.: Geometric structures on 3-manifolds. In: Handbook of Geometric Topology, pp. 93–164. North-Holland, Amsterdam (2002) Google Scholar
- 16.Kapovich, M.: Hyperbolic Manifolds and Discrete Groups: Lectures on Thurston’s Hyperbolization. Birkhäuser’s Series “Progress in Mathematics”. Birkhäuser, Basel (2000) Google Scholar
- 17.Kapovich, M., Kollár, J.: Fundamental groups of links of isolated singularities. Preprint, arXiv:1109.4047 (2011)
- 22.Selberg, A.: On discontinuous groups in higher-dimensional symmetric spaces. In: Contributions to Function Theory, pp. 147–164. Tata Institute of Fundamental Research, Bombay (1960) Google Scholar
- 24.Ushijima, A.: Density of the centers of generic fundamental polyhedra for purely loxodromic Kleinian groups. Preprint, January 2012 Google Scholar