## Volumes of Picard modular surfaces

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**139**(2011), 3045-3056 Request permission

## Abstract:

We show that the conjectural cusped complex hyperbolic 2-orbifolds of minimal volume are the two smallest arithmetic complex hyperbolic 2-orbifolds. We then show that every arithmetic cusped complex hyperbolic 2-manifold of minimal volume covers one of these two orbifolds. We also give all minimal volume manifolds that simultaneously cover both minimal orbifolds.## References

- Roger C. Alperin,
*An elementary account of Selberg’s lemma*, Enseign. Math. (2)**33**(1987), no. 3-4, 269–273. MR**925989** - Mikhail Belolipetsky,
*On volumes of arithmetic quotients of $\textrm {SO}(1,n)$*, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)**3**(2004), no. 4, 749–770. MR**2124587** - Mikhail Belolipetsky and Vincent Emery,
*On volumes of arithmetic quotients of $\textrm {PO}(n, 1)$, $n$ odd*, preprint, 2010. - Armand Borel and Gopal Prasad,
*Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups*, Inst. Hautes Études Sci. Publ. Math.**69**(1989), 119–171. MR**1019963**, DOI 10.1007/BF02698843 - Wieb Bosma, John Cannon, and Catherine Playoust,
*The Magma algebra system. I. The user language*, J. Symbolic Comput.**24**(1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR**1484478**, DOI 10.1006/jsco.1996.0125 - Donald I. Cartwright and Tim Steger,
*Enumeration of the 50 fake projective planes*, C. R. Math. Acad. Sci. Paris**348**(2010), no. 1-2, 11–13 (English, with English and French summaries). MR**2586735**, DOI 10.1016/j.crma.2009.11.016 - Ted Chinburg, Eduardo Friedman, Kerry N. Jones, and Alan W. Reid,
*The arithmetic hyperbolic 3-manifold of smallest volume*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**30**(2001), no. 1, 1–40. MR**1882023** - Marston Conder and Colin Maclachlan,
*Compact hyperbolic 4-manifolds of small volume*, Proc. Amer. Math. Soc.**133**(2005), no. 8, 2469–2476. MR**2138890**, DOI 10.1090/S0002-9939-05-07634-3 - Patrick B. Eberlein,
*Geometry of nonpositively curved manifolds*, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996. MR**1441541** - Elisha Falbel and John R. Parker,
*The geometry of the Eisenstein-Picard modular group*, Duke Math. J.**131**(2006), no. 2, 249–289. MR**2219242**, DOI 10.1215/S0012-7094-06-13123-X - William M. Goldman,
*Complex hyperbolic geometry*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. MR**1695450** - Rolf-Peter Holzapfel,
*Geometry and arithmetic around Euler partial differential equations*, Mathematics and its Applications (East European Series), vol. 11, D. Reidel Publishing Co., Dordrecht, 1986. MR**867406** - Rolf-Peter Holzapfel,
*Ball and surface arithmetics*, Aspects of Mathematics, E29, Friedr. Vieweg & Sohn, Braunschweig, 1998. MR**1685419**, DOI 10.1007/978-3-322-90169-9 - Serge Lang,
*Algebraic number theory*, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR**1282723**, DOI 10.1007/978-1-4612-0853-2 - G. D. Mostow,
*Braids, hypergeometric functions, and lattices*, Bull. Amer. Math. Soc. (N.S.)**16**(1987), no. 2, 225–246. MR**876959**, DOI 10.1090/S0273-0979-1987-15510-8 - John R. Parker,
*On the volumes of cusped, complex hyperbolic manifolds and orbifolds*, Duke Math. J.**94**(1998), no. 3, 433–464. MR**1639519**, DOI 10.1215/S0012-7094-98-09418-2 - Vladimir Platonov and Andrei Rapinchuk,
*Algebraic groups and number theory*, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR**1278263** - Gopal Prasad,
*Volumes of $S$-arithmetic quotients of semi-simple groups*, Inst. Hautes Études Sci. Publ. Math.**69**(1989), 91–117. With an appendix by Moshe Jarden and the author. MR**1019962**, DOI 10.1007/BF02698841 - Gopal Prasad and Sai-Kee Yeung,
*Fake projective planes*, Invent. Math.**168**(2007), no. 2, 321–370. MR**2289867**, DOI 10.1007/s00222-007-0034-5 - Winfried Scharlau,
*Quadratic and Hermitian forms*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985. MR**770063**, DOI 10.1007/978-3-642-69971-9 - William P. Thurston,
*Shapes of polyhedra and triangulations of the sphere*, The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 511–549. MR**1668340**, DOI 10.2140/gtm.1998.1.511 - J. Tits,
*Reductive groups over local fields*, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR**546588** - Hanns Zeltinger,
*Spitzenanzahlen und Volumina Picardscher Modulvarietäten*, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 136, Universität Bonn, Mathematisches Institut, Bonn, 1981 (German). Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn, 1981. MR**645358**

## Additional Information

**Matthew Stover**- Affiliation: Department of Mathematics, University of Texas at Austin, One University Station C1200, Austin, Texas 78712-0257
- Address at time of publication: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109-1043
- MR Author ID: 828977
- Email: mstover@math.utexas.edu
- Received by editor(s): May 30, 2010
- Received by editor(s) in revised form: August 5, 2010
- Published electronically: January 14, 2011
- Communicated by: Alexander N. Dranishnikov
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**139**(2011), 3045-3056 - MSC (2010): Primary 11F06, 22E40, 20G20
- DOI: https://doi.org/10.1090/S0002-9939-2011-10786-X
- MathSciNet review: 2811261