Log-Riemann surfaces, Caratheodory convergence and Euler’s formula

Quantum Statistical Mechanics of the Absolute Galois Group

“…For a Belyi map f : Σ = H/Γ → P 1 = H/∆ the uniformization equation can be seen as the pullback of the equation for H/∆. A formulation of multivalued inverse developing maps of Belyi maps in terms of log-Riemann surfaces is discussed in [7].…”

Uniformization of simply connected finite type Log-Riemann surfaces

“…We first recall some basic properties of log-Riemann surfaces from [BPM10a]. A log-Riemann surface S is equipped with a local holomorphic diffeomorphism π : S → C such that the following holds: the flat metric |dπ| induces a path metric d on S; letting S = S ⊔ R be the metric completion of S, the set R of points added is discrete.…”

“…We now argue as in the proof of Theorem 1.1 of [BPM10b]. Let F : C → S * be a uniformization such that F (0) = q, and choose a basepoint q ′ = q in S. The approximation Theorem 2.10 of [BPM10b] gives a sequence of pointed finite-sheeted log-Riemann surfaces (S k , q ′ k ) converging to (S, q ′ ) in the sense of Caratheodory (see [BPM10a]). For k large the surfaces S × k also have one end which is a degree (−K) covering of {|z| > R}, and hence as above we can add a point q k to obtain a Riemann surface S * k which is biholomorphic to C. Let F k : C → S * k be a uniformization such that F k (0) = q k .…”

“…By the compactness Theorem 2.11 of [BPM10b], there is a subsequence of S N which converges in the sense of Caratheodory to a log-Riemann surface structure S on C * with at most n ramification points. By the Caratheodory convergence Theorem 1.2 of [BPM10a], the maps R N converge along the same subsequence uniformly on compacts to the projection mapping of S, which is hence given by F . It follows that there are at most n points added in the completion of a neighbourhood of ∞ with respect to the metric induced by |η|.…”

“…In [BPM10a] we defined the notion of log-Riemann surface, as a Riemann surface S equipped with a local diffeomorphism π : S → C such that the set of points R added in the completion S * = S ⊔ R of S with respect to the flat metric on S induced by |dπ| is discrete. It was shown in [BPM10a] that π extends to the points p ∈ R, and is a covering of a punctured neighbourhood of p onto a punctured disk in C; the point p is called a ramification point of S of order equal to the degree of the covering π near p. The finite order ramification points may be added to S to give a Riemann surface S × , called the finite completion of S. We call a log-Riemann surfaces of finite type if it has finitely many ramification points and finitely generated fundamental group.…”