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 $n$ –manifold $M$ . These spaces are closed semi-algebraic subsets of the variety of characters of representations of $π_{1} (M)$ in $S L_{n + 1} (ℝ)$ . 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 $π_{1} (M)$ 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 $S L_{n + 1}$ 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.

Keywords

projective structure, Bruhat–Tits building, tropical geometry, character, representation

Mathematical Subject Classification 2000

Primary: 57M60, 57M50, 51E24, 57N16

References

Publication

Received: 26 July 2007

Accepted: 20 November 2007

Published: 12 March 2008

Authors

Daniele Alessandrini
Viale Pola 23 00198 Roma Italy	Dipartimento di Matematica Università di Pisa Italy

Volume 8, issue 1 (2008)

Daniele Alessandrini

Abstract

Keywords

Mathematical Subject Classification 2000

References

Publication

Authors