## Topology & Geometric Group Theory Seminar

## Fall 2011

### 1:30 – 2:30, Malott 203

### Tuesday, November 22

**Harold
Sultan**, Columbia University

*Hyperbolic directions in CAT(0) spaces*

A Morse geodesic is defined by the property that quasi-geodesics with
endpoints on the geodesic remain a bounded distance from the geodesic.
Contracting geodesics on the other hand are defined by the property
that metric balls disjoint from the geodesic have uniformly bounded
nearest point projections onto the geodesic. In hyperbolic space all
geodesics are both Morse and contracting, while in Euclidean space
there are no Morse or contracting geodesics. More generally,
throughout the literature on geometric group theory, one is frequently
interested in understanding families of (quasi-)geodesics which admit
hyperbolic behavior, such as being Morse or contracting. In this
talk, I prove that in CAT(0) spaces a quasi-geodesic is Morse if and
only if it is contracting. In particular, as a corollary this
resolves an open question of Drutu-Mozes-Sapir regarding a converse to
their characterization of Morse quasi-geodesics in the asymptotic
cone.

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