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Kähler groups, \(\mathbb R\)-trees, and holomorphic families of Riemann surfaces. (English) Zbl 1350.53093

Author’s abstract: Let \(X\) be a compact Kähler manifold, and \(g\) a fixed genus. Due to the work of A. N. Parshin [Izv. Akad. Nauk SSSR, Ser. Mat. 32, 1191–1219 (1968; Zbl 0181.23902)] and S. Ju. Arakelov [Math. USSR, Izv. 5, 1277–1302 (1972; Zbl 0248.14004)], it is known that there are only a finite number of non-isotrivial holomorphic families of Riemann surfaces of genus \(g\geq 2\) over \(X\). We prove that this number only depends on the fundamental group of \(X\). Our approach uses geometric group theory (limit groups, \(\mathbb R\)-trees, the asymptotic geometry of the mapping class group), and Gromov-Shoen theory. We prove that in many important cases, limit groups (in the sense of Z. Sela [Publ. Math., Inst. Hautes Étud. Sci. 93, 31–105 (2001; Zbl 1018.20034)]) associated to infinite sequences of actions of a Kähler group on a Gromov-hyperbolic space are surface groups and we apply this result to monodromy groups acting on complexes of curves

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
20F65 Geometric group theory
32Q15 Kähler manifolds
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