## Algebraic & Geometric Topology

### Tropicalization of group representations

Daniele Alessandrini

#### Abstract

In this paper we give an interpretation to the boundary points of the compactification of the parameter space of convex projective structures on an $n$–manifold $M$. These spaces are closed semi-algebraic subsets of the variety of characters of representations of $π1(M)$ in $SLn+1(ℝ)$. 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 $π1(M)$ 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 $SLn+1$ to nonarchimedean fields with real surjective valuation. In the case $n=1$ 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 8, Number 1 (2008), 279-307.

Dates
Accepted: 20 November 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796814

Digital Object Identifier
doi:10.2140/agt.2008.8.279

Mathematical Reviews number (MathSciNet)
MR2443230

Zentralblatt MATH identifier
1170.51005

#### Citation

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

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