Algebraic & Geometric Topology

Tropicalization of group representations

Daniele Alessandrini

Full-text: Open access

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(). 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
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


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References

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