On a theorem of Abikoff
Lipman Bers · Annales Academiae Scientiarum Fennicae Series A I Mathematica · 1985
This note contains a new proof and an extension of a theorem of Abikoff [l] on (complex) boundaries of Teichmüller spaces. First we recall some definitions and results, cf. [3] for references. Let G be a Fuchsian group operating on the upper half-plane (1, i.e., a discrete subgroup of PSL(2, B). Let B(L, G) be the eomplex Banash sBase sf hslomorBhio functions E(z) defined in the lower hatf-plane Z, with norm llEll: suP IYzEQ)l '- (z: x*iY(L), and satisfying the functional equation ofquadratic differentials E(ek))s ' k) ' : Q(z), c(G. For every eeB(L, G) the Schwarzian differential equation(1) {r,z}:(#E)'-+(#&)':QQ) has meromorphic solutions in-L; if W is one, all others are of the form aW wherc a€PSL(2, C). k is convenient to denote bv W, the solution of (l) normalized by the requirement that %(t-t) : +*o$), /--- o. Every EQB(Z, G) induces a homomorphism X, of G into PSL(2, C) defined by the rule IAr " SQ) : Xr(g) "Wr(t) k(G, z(L). (The group G*:x,(A is called the monodromy group of 9.) The Teichmiiller space "(G) of G can be defined as the set of those E(.B(L, q for which \\ isthe restriction to L of a quasiconformal self-map tft, of i, with frrogotftrl(z): Tark)Q) G€G, zee) (so that G*is a quasi-Fuchsian group).