Handlebodies, Outer space, and tropical geometry
Rohini Ramadas, Rob Silversmith, Karen Vogtmann, Rebecca R. Winarski

TL;DR
This paper explores the tropicalization of moduli spaces of Riemann surfaces and handlebodies, connecting complex geometry, tropical geometry, and geometric group theory through new constructions and generalizations.
Contribution
It introduces the notion of stable complex handlebodies and shows how certain complex moduli spaces relate to their tropical counterparts, extending known relationships to punctured cases.
Findings
CV_{g,n}^* as tropicalization of complex handlebody moduli space
Construction of a simply connected partial compactification with normal crossings
Extension of relationships between geometric group theory objects to punctured surfaces
Abstract
The moduli space of graphs is a polyhedral object that mimics the behavior of the moduli spaces , of (stable) Riemann surfaces; this relationship has been made precise in several different ways, which collectively identify as the "tropicalization" of . We describe how this relationship lifts to some objects that live over (like Teichm\"uller space) and that live over (like the Culler-Vogtmann space ). We introduce the notion of a stable complex handlebody, and show that can be viewed as the tropicalization of a certain complex manifold that parametrizes complex handlebodies. An important ingredient is our construction of a partial compactification , which we prove is a simply…
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