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In this article, we study various concrete algebraic and differential geometric properties of the Cartwright-Steger surface, the unique smooth surface of Euler number 3 which is neither a projective plane nor a fake projective plane. In particular, we determine the genus of a generic fiber of the Albanese fibration, and deduce that the singular fibers are not totally geodesic, answering an open problem about fibrations of a complex ball quotient over a Riemann surface.The Main Theorem follows from Theorem 3, Corollary 2, Lemma 12, Corollary 1, Lemma 8 and Proposition 12. As an immediate consequence, see Theorem 4, we have answered an open problem communicated to us by Ngaiming Mok on properties of fibrations on complex ball quotients.Corollary There exists a fibration of a smooth complex two ball quotient over a smooth Riemann surface with non-totally geodesic singular fibers.Here are a few words about the presentation of the article. To streamline our arguments and to make the results more understandable, we state and prove the geometric results of the article sequentially in the sections 3 to 5 of the article. Many of the results rely on computations in the groups Π andΓ, often obtained with assistance of the algebra package Magma, and we present these exclusively in the first two sections of the paper (except for the proof of Proposition 12) with a geometric perspective each time it is possible. More details appear in a longer version of this paper and on the webpage of the first author [CKY].Acknowledgments. Donald Cartwright thanks Jonathan Hillman for help in obtaining standard presentations of surface groups from non-standard ones. Vincent Koziarz would like to thank Aurel Page for his help in computations involving Magma, Duc-Manh Nguyen for very useful conversations about triangle groups and Riemann surfaces, Arnaud Chéritat for his help in drawing pictures and Frédéric Campana for useful comments. Sai-Kee Yeung would like to thank Martin Deraux, Igor Dolgachev, Ching-Jui Lai, Ngaiming Mok and Domingo Toledo for their interest and helpful comments. The main results of this paper were presented at the 4th South Kyushu workshop on algebra, complex ball quotients and related topics, July 22-25, 2014, Kumamoto, Japan. The first and the third authors thank Fumiharu Kato for his kind invitation.Since completing this paper, we were informed by Domingo Toledo that he, Fabrizio Catanese, JongHae Keum and Matthew Stover had independently proved some of our results in a paper they are preparing.

In this note we show that, for any log-canonical pair (X, ∆), K X + ∆ is Q-effective if its Chern class contains an effective Q-divisor.

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Let ρ be a maximal representation of a uniform lattice Γ ⊂ SU(n, 1), n ≥ 2, in a classical Lie group of Hermitian type G. We prove that necessarily G = SU(p, q) with p ≥ qn and there exists a holomorphic or antiholomorphic ρ-equivariant map from the complex hyperbolic space to the symmetric space associated to SU(p, q). This map is moreover a totally geodesic homothetic embedding. In particular, up to a representation in a compact subgroup of SU(p, q), the representation ρ extends to a representation of SU(n, 1) in SU(p, q).