Tropical Teichmüller and Siegel spaces (Q2867773)

This paper recasts some well-known results and constructions from geometric group theory in the context of (combinatorial) tropical geometry -- specifically, by viewing outer space as a sort of tropical Teichmüller space and its quotient by the mapping class group, the moduli space of metric graphs, as the moduli space of tropical curves -- and then introduces a similar construction for abelian varieties, namely, to introduce a sort of tropical Siegel space whose quotient by the integral general linear group (thought of as a tropical symplectic group) yields the tropical moduli space of principally polarized abelian varieties (or more accurately, this Siegel space depends on a combinatorial choice corresponding to the different toroidal compactifications of its quotient). The culmination is then a tropical period map that descends to yield the previously studied tropical Torelli map(s).NEWLINENEWLINEThere is some ambiguity over what in this paper had already been done by the geometric group theorists, but I believe the point in this paper is to collect the results, both old and new, that might interest a tropical geometer, and to spell out all the necessary constructions explicitly in a language of fans that conforms to one of the current (quite combinatorial) approaches to tropical geometry. Thus one has here a toolkit allowing for a convenient approach to study and prove results about the tropical moduli space of curves, abelian varieties, and their relation, which is easily readable by the intended audience. In fact, that the very recent ideas and constructions of tropical geometry as some kind of extreme limit of algebraic geometry fits so nicely with these ideas from topology and geometric group theory is quite beautiful and satisfying and should be thought to reinforce these subjects. Quite likely much of the work here can subsequently be recast in the formalism introduced by Abramovich-Caporaso-Payne (who studied the moduli space of curves) by studying the non-Archimedean aspects of Teichmüller and Siegel space, but the current paper does a nice job of launching tropical geometers in this healthy direction and reinforcing the bridge to topology/geometric group theory.NEWLINENEWLINEFor the entire collection see [Zbl 1266.14003].