Tropical Teichmuller and Siegel spaces

TL;DR

This paper explores the connections between tropical moduli spaces, Outer space, and quadratic forms, proposing tropical analogues of classical Teichmüller and Siegel spaces within a unified geometric framework.

Contribution

It introduces a unified categorical framework for tropical moduli spaces and establishes analogies with classical spaces like Teichmüller and Siegel spaces using ideal stacky fans.

Findings

Unified description of tropical moduli spaces and Outer space.

Establishment of period maps linking these spaces.

Proposed tropical analogues of classical geometric spaces.

Abstract

In this paper, we present a unified study of the moduli space of tropical curves and Outer space which we link via period maps to the moduli space of tropical abelian varieties and the space of positive definite quadratic forms. Our work is a first step towards exhibiting Outer space and the space of positive definite quadratic forms as analogues of Teichm\"uller space and Siegel space, respectively, in tropical geometry. All these spaces and the maps among them are described within the category of ideal stacky fans, which we describe in detail.