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On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces with two specified points. (English) Zbl 1026.30037
Let \(S\) be a hyperbolic Riemann surfaces \(R\) of analytically finite type with two specific points \(p_1,p_2\in S\), and set \(\dot S:=S\setminus \{p_1,p_2\}\). Let \(I sot(S,2)\) be the group of orientation preserving homeomorphisms of \(S\) onto itself isotopic to \(id_S\) and fixing the \(p_j\) factored by the normal subgroup of homeomorphisms of \(S\) isotopic to the identity of \(\dot S\). Elements \([\omega]\in I sot(S,2)\) induce canonically elements \(\langle\omega |_S\rangle\) of the Teichmüller modular group \(\text{Mod} (\dot S)\). L. Bers [Acta Math. 141, 73-98 (1978; Zbl 0389.30018)] classified elements of \(\text{Mod}(\dot S)\) as elliptic, parabolic and elliptic using the Teichmüller distance on the Teichmüller space \(T(\dot S)\). In this paper the corresponding classification of elements \([\omega]\) of \(I sot(S,2)\) is described using the strings of the induced pure braids \([b_\omega]\). The results are motivated by a theorem of I. Kra for surfaces with one specific point [Acta Math. 146, 231-270 (1981; Zbl 0477.32024)].

30F10 Compact Riemann surfaces and uniformization