Volume 8, issue 1 (2008)

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Tropicalization of group representations

Daniele Alessandrini

Algebraic & Geometric Topology 8 (2008) 279–307

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.

projective structure, Bruhat–Tits building, tropical geometry, character, representation
Mathematical Subject Classification 2000
Primary: 57M60, 57M50, 51E24, 57N16
Received: 26 July 2007
Accepted: 20 November 2007
Published: 12 March 2008
Daniele Alessandrini
Viale Pola 23
00198 Roma
Dipartimento di Matematica
Università di Pisa