The space of extended Schottky groups
Let M be a group of Möbius transformations of the Riemann sphere \(\Sigma\). A subgroup \(\Gamma\) of M is said to be an extended Schottky group of type (g,s,m) with a standard system of generators \(T_ i\), \(W_ j\), \(V_ k\), \(U_ k\) and determining curves \(C_ i\), \(C_ i'\), \(B_ j\), \(B_ j'\), \(L_ k\), where \(L_ k\) is a topological quadrangle with the edges \(K_ k\), \(K_ k'\), \(P_ k\), \(P_ k'\), provided that 1) \(B_ j\cap B_ j'\) is a point \(p_ j\) and mutually disjoint curves \(C_ i\), \(C_ i'\), \(B_ j\cup B_ j'\), \(L_ k\), \(i=1,...,g\); \(j=1,...,s\); \(k=1,...,m\), form the boundary of a \(2g+s+m\)-connected region D, 2) \(T_ i\) is a loxodromic element and \(T_ i(C_ i)=C_ i'\), \(T_ i(D)\cap D=\emptyset\), \(W_ j\) is a parabolic element and \(W_ j(B_ j)=B_ j'\), \(W(p_ j)=p_ j\), \(W_ j(D)\cap D=\emptyset\) the free commutative group \(<V_ k,U_ k>\) contains only parabolic elements and \(U_ k(K_ k)=K_ k'\), \(V_ k(P_ k)=P_ k'\), \(U_ k(D)\cap D=\emptyset\), \(V_ k(D)\cap D=\emptyset.\) In this paper the authors give an algebraic characterization of extended Schottky groups of type (g,s,m) (Theorem 1). One can naturally introduce the space of extended Schottky groups \(S_{(g,s,m)}\). This space can be embedded as an open connected manifold \(S^*_{(g,s,m)}\) in \(C^{3g+3m+2s-3}\). This allows to consider its boundary \(\partial S^*_{(g,s,m)}\). Further results (theorems 2,3,4) give a characterization of the points of \(\partial S^*_{(g,s,m)}\).
scientific article; zbMATH DE number 4006636
scientific article