Yang, Qingjie Matrix classes realized by automorphisms of compact Riemann surfaces of genus two. (English) Zbl 1254.57017 Commun. Algebra 40, No. 7, 2343-2357 (2012). It is known that if \(S\) is a compact Riemann surface of genus \(g > 1\) with (conformal) automorphism group \(G\), then there exists a faithful group homomorphism from \(G =\) Aut\((S)\) into the symplectic group Sp\({}_{2g}({\mathbb Z})\), given by the natural action of \(G\) as a group of automorphisms of the first homology group \(H_1(S,{\mathbb Z})\). In this paper, which in some sense is a sequel to a 1997 paper by D. Sjerve and the author [J. Algebra 195, No. 2, 580–603 (1997; Zbl 0888.20024)], a complete list is given of representatives of all conjugacy classes of torsion elements in Sp\({}_4({\mathbb Z})\) that can occur as images of elements of \(G\) under such a monomorphism in the case \(g = 2\). The motivation for doing this is not entirely clear, but the paper appears to be correct and well written. Reviewer: Marston Conder (Auckland) MSC: 57M60 Group actions on manifolds and cell complexes in low dimensions 20G99 Linear algebraic groups and related topics 30F10 Compact Riemann surfaces and uniformization Keywords:automorphism; compact Riemann surface; first homology group; symplectic group Citations:Zbl 0888.20024 PDFBibTeX XMLCite \textit{Q. Yang}, Commun. Algebra 40, No. 7, 2343--2357 (2012; Zbl 1254.57017) Full Text: DOI References: [1] Aigon , A. ( 2001 ).Transformations Hyperboliques et Courbes Algébriques en Genres 2 et 3.Ph.D. thesis. Johns Hopkins University Press . [2] Bolze , O. ( 1887 ). On Binary Sextics with Linear Transformations in Themselves . [3] Eckmann B., Invent. Math. 69 pp 293– (1982) · Zbl 0501.20031 · doi:10.1007/BF01399508 [4] Farkas H. M., Riemann Surfaces. 71, 2. ed. (1992) · doi:10.1007/978-1-4612-2034-3 [5] Gabai D., Ann. Math. 136 pp 447– (1992) · Zbl 0785.57004 · doi:10.2307/2946597 [6] Harvey W. J., Quart. J. Math. Oxford 17 (2) pp 86– (1966) · Zbl 0156.08901 · doi:10.1093/qmath/17.1.86 [7] Harvey W. J., Discrete Groups and Automorphic Functions (1977) [8] Jones G., Complex Functions (1987) [9] Kerckhoff S. P., Ann. Math. 117 pp 235– (1983) · Zbl 0528.57008 · doi:10.2307/2007076 [10] MacBeath A. M., Bull. London Math. Soc. 5 pp 103– (1973) · Zbl 0259.30016 · doi:10.1112/blms/5.1.103 [11] Nielsen J., Acta Math. 75 pp 23– (1943) · Zbl 0027.26601 · doi:10.1007/BF02404101 [12] Sjerve D., J. Algebra 195 (2) pp 580– (1997) · Zbl 0888.20024 · doi:10.1006/jabr.1997.7084 [13] Vick J. W., Homology Theory (1973) [14] Wiman A., Bihang Till. Kongl. Svenska Vetenskaps Akademiens Handlingar, bd. 21 pp 1– (1985) [15] Yang Q., J. Math. Research & Exposition 28 (1) pp 177– (2008) 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.