The infinite Nielsen kernels of some bordered Riemann surfaces (Q1330152)

The author considers an interesting problem which is probably not as well known as it should be. If \(X\) is a finite bordered Riemann surface, not a doubly-connected domain (there is to be at least one border component) then for every border component \(C\) there is a closed geodesic \(C'\) in the Poincaré metric which with \(C\) bounds a doubly-connected subdomain of \(X\). The \(C'\) for various \(C\) are disjoint and bound a finite bordered Riemann surface \(N'(x)\) called the Nielsen kernel of \(X\). If one forms the successive Nielsen kernels \(N^{k+1}(x)= N'\) \((N^ k(x))\), \(k= 1,2,\dots\), the problem, formulated by \textit{L. Bers} [Ann. Acad. Sci. Fenn., Ser. AI 2, 29-34 (1976; Zbl 0352.30014)], is to investigate the intersection \(\bigcap^ \infty_{k=1} N^ k(x)\), called the infinite Nielsen kernel \(N^ \infty(x)\) of \(X\). Actually in this paper the author considers the generalized situation for orbifolds which are here Riemann surfaces with a discrete set of distinguished points to which integral indexes \((>1)\) are attached. He gives a basic characterization for the infinite Nielsen kernel and a proposition which relates the infinite Nielsen kernel of an orbifold with that of a covering of finite constant valence. He shows how these can be used to determine the infinite Nielsen kernels for a nubmer of examples.