Tropicalization of group representations

Alessandrini, Daniele¹

¹Viale Pola 23, 00198 Roma, Italy, Dipartimento di Matematica, Università di Pisa, Italy

Algebraic and Geometric Topology, Tome 8 (2008) no. 1, pp. 279-307

Résumé

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.

Détail

DOI : 10.2140/agt.2008.8.279

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

Bibliographie

[1] D Alessandrini, A compactification for the spaces of convex projective structures on manifolds

[2] D Alessandrini, Logarithmic limit sets of real semi-algebraic sets, submitted for publication

[3] G Cohen, S Gaubert, J P Quadrat, Duality and separation theorems in idempotent semimodules, Linear Algebra Appl. 379 (2004) 395

[4] M Joswig, B Sturmfels, J Yu, Affine buildings and tropical convexity

[5] I Kim, Rigidity and deformation spaces of strictly convex real projective structures on compact manifolds, J. Differential Geom. 58 (2001) 189

[6] J W Morgan, P B Shalen, Valuations, trees, and degenerations of hyperbolic structures I, Ann. of Math. $(2)$ 120 (1984) 401

[7] J W Morgan, P B Shalen, Degenerations of hyperbolic structures II: Measured laminations in 3–manifolds, Ann. of Math. $(2)$ 127 (1988) 403

[8] J W Morgan, P B Shalen, Degenerations of hyperbolic structures III: Actions of 3–manifold groups on trees and Thurston's compactness theorem, Ann. of Math. $(2)$ 127 (1988) 457

[9] T Ohtsuki, Problems on invariants of knots and 3-manifolds, from: "Invariants of knots and 3-manifolds (Kyoto, 2001)" (editors T Ohtsuki, T Kohno, T Le, J Murakami, J Roberts, V Turaev), Geom. Topol. Monogr. 4, Geom. Topol. Publ., Coventry (2002) 377