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 Wequivariant 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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In this paper, we investigate the set of accumulation points of normalized roots of infinite Coxeter groups for certain class of their action. Concretely, we prove the conjecture proposed in [6, Section 3.2] in the case where the equipped Coxeter matrices are of type (n − 1, 1), where n is the rank. Moreover, we obtain that the set of such accumulation points coincides with the closure of the orbit of one point of normalized limit roots. In addition, in order to prove our main results, we also investigate some properties on fixed points of the action.

In this paper, we give an alternative approach to Royden–Earle–Kra–Markovic's characterization of biholomorphic automorphisms of Teichmüller space of Riemann surface of analytically finite type.

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