Analogues of the elliptic modular functions in R³

Summary

Introduction

The purpose of this talk is to present mappings g of the upper half space H = {X ∈ R³: x₃ > 0 } into R³ which resemble in some respects the elliptic modular function and raise some questions in the theory of quasiregular mappings. Each of the mappings g has the following properties:

(1) g is continuous, discrete, open, sense-preserving and has a bounded dilatation in H.

(2) g defines a closed map of H onto R³\P for some set P ⊂ R³ of finite cardinality.

(3) there exists a discrete group G of Mobius transformations acting on H with a non-compact fundamental domain of finite hyperbolic volume in H such that g° A = g for all A ∈ G.

(4) g has no limit at any point b ∈ ∂H.

(1) says that g is quasiregular in the sense of Martio, Rickman and Väisäiä[l]. This means that g ∈ ACL³ and |g'(x)|³ < K(x, g) a. e. in H for some K ∈ [1, ∞), where |g'(x)| denotes the sup norm of the formal derivative g'(x), and J(x, g) = det g'(x).

The class of quasiregular mappings in R³ is a reasonable generalization of holomorphic functions in C; and with (2)–(4) we may consider the mappings g as analogues of the elliptic modular function. However, contrary to the elliptic modular function, none of the mappings g that are constructed here is a local homeomorphism. Martio and I show in [4] that no quasiregular mapping in Rⁿ, n > 3, which satisfies (3) is a local homeomorphism.