Dirichlet fundamental domains and topology of projective varieties. (English) Zbl 1348.20045
Summary: 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.
MSC:
20F65 | Geometric group theory |
57M05 | Fundamental group, presentations, free differential calculus |
14B05 | Singularities in algebraic geometry |
14J17 | Singularities of surfaces or higher-dimensional varieties |
14F35 | Homotopy theory and fundamental groups in algebraic geometry |
Keywords:
finitely presented groups; irreducible complex projective varieties; fundamental groups; singularities; normal crossings; Whitney umbrellasReferences:
[1] | Amorós, J., Burger, M., Corlette, K., Kotschick, D., Toledo, D.: Fundamental Groups of Compact Kähler Manifolds. Mathematical Surveys and Monographs, vol. 44. Am. Math. Soc., Providence (1996) · Zbl 0849.32006 · doi:10.1090/surv/044 |
[2] | Armstrong, M.: The fundamental group of the orbit space of a discontinuous group. Math. Proc. Camb. Philos. Soc. 64, 299-301 (1968) · Zbl 0159.53002 · doi:10.1017/S0305004100042845 |
[3] | Atkinson, M.D.: Primitive spaces of matrices of bounded rank. II. J. Aust. Math. Soc. A 34(3), 306-315 (1983) · Zbl 0521.15009 · doi:10.1017/S1446788700023740 |
[4] | Beardon, A., Maskit, B.: Limit points of Kleinian groups and finite-sided fundamental polyhedra. Acta Math. 132, 1-12 (1974) · Zbl 0277.30017 · doi:10.1007/BF02392106 |
[5] | Bonahon, F., Geometric structures on 3-manifolds, 93-164 (2002), Amsterdam · Zbl 0997.57032 |
[6] | Bowditch, B.: Geometrical finiteness for hyperbolic groups. J. Funct. Anal. 113(2), 245-317 (1993) · Zbl 0789.57007 · doi:10.1006/jfan.1993.1052 |
[7] | Brown, K.: Cohomology of Groups. Springer, Berlin (1982) · Zbl 0584.20036 · doi:10.1007/978-1-4684-9327-6 |
[8] | Carlson, J., Toledo, D.: Harmonic mappings of Kähler manifolds to locally symmetric spaces. Publ. Math. IHÉS 69, 173-201 (1989) · Zbl 0695.58010 · doi:10.1007/BF02698844 |
[9] | Diaz, R., Ushijima, A.: On the properness of some algebraic equations appearing in Fuchsian groups. Topol. Proc. 33, 81-106 (2009) · Zbl 1179.30039 |
[10] | Drumm, T., Poritz, J.: Ford and Dirichlet fundamental domains for cyclic subgroups of \(\operatorname{\mathit{PSL}}(2,\mathbb{C})\) acting on ℍ3 and ∂ℍ3. Conform. Geom. Dyn. 3, 116-150 (1999) · Zbl 0999.30028 · doi:10.1090/S1088-4173-99-00042-9 |
[11] | Eisenbud, D., Harris, J.: Vector spaces of matrices of low rank. Adv. Math. 70(2), 135-155 (1988) · Zbl 0657.15013 · doi:10.1016/0001-8708(88)90054-0 |
[12] | Goresky, M., MacPherson, R.: Stratified Morse Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 14. Springer, Berlin (1988) · Zbl 0639.14012 · doi:10.1007/978-3-642-71714-7 |
[13] | Haefliger, A., Orbi-espaces, No. 83, 203-213 (1990), Basel |
[14] | Harris, J.: Algebraic Geometry: A First Course. Graduate Texts in Mathematics, vol. 133. Springer, Berlin (1992) · Zbl 0779.14001 |
[15] | Jorgensen, T.; Marden, A., Generic fundamental polyhedra for Kleinian groups, No. 11, 69-85 (1988), New York · Zbl 0673.30034 · doi:10.1007/978-1-4613-9611-6_7 |
[16] | Kapovich, M.: Hyperbolic Manifolds and Discrete Groups: Lectures on Thurston’s Hyperbolization. Birkhäuser’s Series “Progress in Mathematics”. Birkhäuser, Basel (2000) |
[17] | Kapovich, M., Kollár, J.: Fundamental groups of links of isolated singularities. Preprint, arXiv:1109.4047 (2011) · Zbl 1307.14005 |
[18] | Lovasz, L.: Singular spaces of matrices and their application in combinatorics. Bol. Soc. Bras. Mat. 20(1), 87-99 (1989) · Zbl 0757.05035 · doi:10.1007/BF02585470 |
[19] | Maskit, B.: Kleinian Groups. Springer, Berlin (1988) · Zbl 0627.30039 |
[20] | Panov, D., Petrunin, A.: Telescopic actions. Geom. Funct. Anal. 22(6), 1814-1831 (2012) · Zbl 1271.57051 · doi:10.1007/s00039-012-0194-3 |
[21] | Ratcliffe, J.: Foundations of Hyperbolic Manifolds, 2nd edn. Graduate Texts in Mathematics, vol. 149. Springer, New York (2006) · Zbl 1106.51009 |
[22] | Selberg, A., On discontinuous groups in higher-dimensional symmetric spaces, 147-164 (1960), Bombay · Zbl 0201.36603 |
[23] | Simpson, C.: Local systems on proper algebraic V-manifolds. Pure Appl. Math. Q. 7, 1675-1760 (2011) · Zbl 1316.14008 · doi:10.4310/PAMQ.2011.v7.n4.a27 |
[24] | Ushijima, A.: Density of the centers of generic fundamental polyhedra for purely loxodromic Kleinian groups. Preprint, January 2012 · Zbl 0277.30017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.