Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 8, Number 1 (2008), 279-307.
Tropicalization of group representations
In this paper we give an interpretation to the boundary points of the compactification of the parameter space of convex projective structures on an –manifold . These spaces are closed semi-algebraic subsets of the variety of characters of representations of in . The boundary was constructed as the “tropicalization” of this semi-algebraic set. Here we show that the geometric interpretation for the points of the boundary can be constructed searching for a tropical analogue to an action of on a projective space. To do this we need to construct a tropical projective space with many invertible projective maps. We achieve this using a generalization of the Bruhat–Tits buildings for to nonarchimedean fields with real surjective valuation. In the case these objects are the real trees used by Morgan and Shalen to describe the boundary points for the Teichmüller spaces. In the general case they are contractible metric spaces with a structure of tropical projective spaces.
Algebr. Geom. Topol., Volume 8, Number 1 (2008), 279-307.
Received: 26 July 2007
Accepted: 20 November 2007
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 57M60: Group actions in low dimensions 57M50: Geometric structures on low-dimensional manifolds 51E24: Buildings and the geometry of diagrams 57N16: Geometric structures on manifolds [See also 57M50]
Alessandrini, Daniele. Tropicalization of group representations. Algebr. Geom. Topol. 8 (2008), no. 1, 279--307. doi:10.2140/agt.2008.8.279. https://projecteuclid.org/euclid.agt/1513796814