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Quasiconformal homogeneity of hyperbolic surfaces with fixed-point full automorphisms. (English) Zbl 1133.30012
The main result is the following
Theorem. For each \(c\in (0,2]\), there exists \(K_c> 1\), such that if \(S\) is a \(K\)-quasiconformal homogeneous closed hyperbolic surface of genus \(g\) that admits a non-trivial conformal automorphism with at least \(c(g+ 1)\) fixed points, then \(K\geq K_c\).
The authors consider the strongly (extremely) \(K\)-quasiconformally homogeneous hyperbolic surface as a surface \(S\), such that for any \(x,y\in S\), there is a \(K\)-quasiconformal homeomorphism of taking \(x\) to \(y\), which is homotopic to a conformal automorphism (identity) of \(S\).
In these cases, one can bound the associated quasiconformal homogeneity constant uniformly away from 1.
Reviewer: A. Neagu (Iaşi)

MSC:
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
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