Volume 10, issue 4 (2006)

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Universal circles for quasigeodesic flows

Danny Calegari

Geometry & Topology 10 (2006) 2271–2298
 arXiv: math.GT/0406040
Abstract

We show that if $M$ is a hyperbolic $3$–manifold which admits a quasigeodesic flow, then ${\pi }_{1}\left(M\right)$ acts faithfully on a universal circle by homeomorphisms, and preserves a pair of invariant laminations of this circle. As a corollary, we show that the Thurston norm can be characterized by quasigeodesic flows, thereby generalizing a theorem of Mosher, and we give the first example of a closed hyperbolic $3$–manifold without a quasigeodesic flow, answering a long-standing question of Thurston.

Keywords
quasigeodesic flows, universal circles, laminations, Thurston norm, 3-manifolds
Mathematical Subject Classification 2000
Primary: 57R30
Secondary: 37C10, 37D40, 53C23, 57M50