We demonstrate that the question whether or not a given topological ramified covering map of the 2-sphere is Thurston equivalent to a rational map is algorithmically decidable.

Proof: Writing Y (Γ) = Γ\H, it is natural to ask (as we did in the “parabolic” case C/Λ) which elements σ ∈ PSL2(R) = Aut(H) descend to give automorphisms of Y (Γ). The condition we need is clearly that for all z ∈ H and γ ∈ Γ, there exists γ′ ∈ Γ such that σ(γz) = γ′(σz). Of course this holds for all z ⇐⇒ σγ = γ′σ; in other words the condition is that σΓσ−1 = Γ, i.e., that σ ∈ N(Γ). This defin...

We shall give, in an optimal form, a sufficient numerical condition for the finiteness of the fundamental group of the smooth locus of a normal K3 surface. We shall moreover prove that, if the normal K3 surface is elliptic and the above fundamental group is not finite, then there is a finite covering which is a complex torus.

Can folding a piece of paper flat make it larger? We explore whether a shape S must be scaled to cover a flat-folded copy of itself. We consider both single folds and arbitrary folds (continuous piecewise isometries S → R2). The underlying problem is motivated by computational origami, and is related to other covering and fixturing problems, such as Lebesgue’s universal cover problem and force ...

We show that the universal cover of a compact complex surface X is the bidisk H × H, or X is biholomorphic to P × P, if and only if K X > 0 and there exists an invertible sheaf η such that η 2 ∼= OX and H0(X,S2Ω1X(−KX) ⊗ η) 6= 0. The two cases are distinguished by the second plurigenus, P2(X) ≥ 2 in the former case, P2(X) = 0 in the latter. We also discuss related questions.

Macbeath gave a formula for the number of xed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the xed-point set of each non-identity element of a cyclic group of automorphisms acting on a closed non-orientable sur...

Let M̃ → M be a holomorphic (unbranched) covering map between two compact complex manifolds, with b2(M̃) = 1. We prove that if M̃ and M both admit regular Kähler forms ω̃ and ω respectively then, up to homotheties, (M̃, ω̃) and (M, ω) are biholomorphically isometric.

We study the topology of codimension one taut foliations of closed orientable 3-manifolds which are smooth along the leaves. In particular, we focus on the lifts of these foliations to the universal cover, specifically when any set of leaves corresponding to nonseparable points in the leaf space can be totally ordered. We use the structure of branching in the lifted foliation to find conditions...

We consider locally symmetric manifolds with a fixed universal covering, and construct for each such manifold M a simplicial complex R whose size is proportional to the volume of M . When M is non-compact, R is homotopically equivalent to M , while when M is compact, R is homotopically equivalent to M \ N , where N is a finite union of submanifolds of fairly smaller dimensions. This expresses h...

In this paper we shall study homomorphisms p: W — Y on minimal transformation groups. We shall prove, in the case that W and Y are metrizable, that W is a finite (iV-to-1) extension of Y if and only if W is an Nfold covering space of Y and pisa covering map. This result places no further restrictions on the acting group. We shall then use this characterization to investigate the question of lif...