Cannon-Thurston maps for Coxeter groups with signature (n-1,1)

2013/12/11 by Ryosuke Mineyama, Mineyama, Ryosuke
Mathematics · #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1312.3174

Abstract

For a Coxeter group W we have an associating bi-linear form B on suitable real vector space. We assume that B has the signature (n-1,1) and all the bi-linear form associating rank n' (≥ 3) Coxeter subgroups generated by subsets of S has the signature (n',0) or (n'-1,1). Under these assumptions, we see that there exists the Cannon-Thurston map for W, that is, the W-equivariant continuous surjection from the Gromov boundary of W to the limit set of W. To see this we construct an isometric action of W on an ellipsoid with the Hilbert metric. As a consequence, we see that the limit set of W coincides with the set of accumulation points of roots of W.

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