##
Geometry of the Complex of Curves II: Hierarchical structure.

### Howard Masur and Yair Minsky

### Abstract

This paper continues a geometric study of Harvey's Complex of Curves,
whose ultimate goal is to apply the theory of hyperbolic spaces and
groups to algorithmic questions for the Mapping Class Group and
geometric properties of Kleinian representations. The authors'
previous result
that the complex is delta-hyperbolic was hard to apply
because the complex is not locally finite; in this paper some tools
are developed for overcoming this problem, and a combinatorial
mechanism introduced which describes sequences of elementary moves in
the graph of markings on a surface. These tools are applied to give a
family of quasi-geodesic words in the Mapping Class Group, and
a linear bound on the shortest word conjugating two
conjugate pseudo-Anosov elements.
A basic tool in the analysis is a family of *subsurface
projections,* which are roughly analogous to closest-point
projections to horoballs in classical hyperbolic space. These
projections have a strong contraction property which makes it possible
to tie together the geometry of the complex and that of the
(infinite) subcomplexes that arise as links of vertices.
The resulting layered structure of the complex is controlled by means
of a combinatorial device called a *hierarchy of geodesics,* which
is the central construction of the paper.

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