Moduli spaces of complex affine and dilation surfaces

TL;DR

This paper constructs and analyzes the structure of moduli spaces for complex affine and dilation surfaces, revealing their geometric and topological properties and their relation to Riemann surface moduli spaces.

Contribution

It introduces the construction of moduli spaces for complex affine and dilation surfaces and describes their geometric structures as bundles and coverings over Riemann surface moduli spaces.

Findings

Moduli space of affine surfaces is a holomorphic affine bundle over Riemann surfaces.

Moduli space of dilation surfaces is a covering space of Riemann surfaces.

Connected components of dilation surface moduli space are orbifold K(G,1).

Abstract

We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech, we show that the the moduli space of affine surfaces with fixed genus and with cone points of fixed complex order is a holomorphic affine bundle over the moduli space of Riemann surfaces. Similarly, the moduli space of dilation surfaces is a covering space of the moduli space of Riemann surfaces. We classify the connected components of the moduli space of dilation surfaces and show that any component is an orbifold K(G,1) where G is the framed mapping class group of Calderon-Salter.