## Algebraic & Geometric Topology

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

### Tropicalization of group representations

#### 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 ${\pi}_{1}\left(M\right)$ in $S{L}_{n+1}\left(\mathbb{R}\right)$. 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 ${\pi}_{1}\left(M\right)$ 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 $S{L}_{n+1}$ 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**

Received: 26 July 2007

Accepted: 20 November 2007

First available in Project Euclid: 20 December 2017

**Permanent link to this document**

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

**Subjects**

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]

**Keywords**

projective structure Bruhat–Tits building tropical geometry character representation

#### 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