Volume 18, issue 4 (2018)

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Algebraic ending laminations and quasiconvexity

Mahan Mj and Kasra Rafi

Algebraic & Geometric Topology 18 (2018) 1883–1916
Abstract

We explicate a number of notions of algebraic laminations existing in the literature, particularly in the context of an exact sequence

$1\to H\to G\to Q\to 1$

of hyperbolic groups. These laminations arise in different contexts: existence of Cannon–Thurston maps; closed geodesics exiting ends of manifolds; dual to actions on $ℝ\phantom{\rule{0.3em}{0ex}}$–trees.

We use the relationship between these laminations to prove quasiconvexity results for finitely generated infinite-index subgroups of $H$, the normal subgroup in the exact sequence above.

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Keywords
hyperbolic group, mapping torus, quasiconvexity, ending lamination, Cannon-Thurston map
Mathematical Subject Classification 2010
Primary: 20F65, 20F67
Secondary: 30F60