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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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