Trees, dendrites and the Cannon–Thurston map

Field, Elizabeth¹

¹Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, United States

Algebraic and Geometric Topology, Tome 20 (2020) no. 6, pp. 3083-3126

Résumé

When 1 → H → G → Q → 1 is a short exact sequence of three word-hyperbolic groups, Mahan Mj (formerly Mitra) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon–Thurston map. In this context, Mj associates to every point z in the Gromov boundary of Q an “ending lamination” on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich and Lustig and Dowdall, Kapovich and Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(FN), one can identify the resultant quotient space with a certain ℝ–tree in the boundary of Culler and Vogtmann’s Outer space.

Détail

Classification : 20F65, 20E07, 20F67, 57M07
Keywords: Cannon–Thurston map, hyperbolic group, algebraic lamination, dendrite, Gromov boundary

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