McShane, Greg (1991) A remarkable identity for lengths of curves. PhD thesis, University of Warwick.

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Abstract

In this thesis we will prove the following new identity

Σγ 1/(1 + exp |γ|) = 1/2,

where the sum is over all closed simple geodesics γ on a punctured torus with
a complete hyperbolic structure, and |γ| is the length of γ. Although it is
well known that there are relations between the lengths of simple geodesics on
a hyperbolic surface (for example the Fricke trace relations and the Selberg
trace formula) this identity is of a wholly different character to anything in
the literature. Our methods are purely geometric; that is, the techniques are
based upon the work of Thurston and others on geodesic laminations rather
than the analytic approach of Selberg.
The first chapter is intended as an exposition of some relevant theory concerning
laminations on a punctured surface. Most important of these results
is that a leaf of a compact lamination cannot penetrate too deeply into a cusp
region. Explicit bounds for the maximum depth are given; in the case of a
torus a simple geodesic is disjoint from any cusp region whose bounding curve
has length less than 4, and this bound is sharp. Another significant result is
that a simple geodesic which enters a small cusp region is perpendicular to
the horocyclic foliation of the cusp region.
The second chapter is concerned with Gcusp, the set of ends of simple
geodesics with at least one end up the cusp. A natural metric on Gcusp
is introduced
so that we can discuss approximation theory. We divide the geodesics
of Gcusp into three classes according to the behaviour of their ends; each class
also has a characterisation in terms of how well any member geodesic can be
approximated. An example is given to demonstrate how this classification generalises
some ideas in the classical theory of Diophantine approximation. The
first class consists of geodesics with both ends up the cusp. Restricting to the
punctured torus it is shown that for such a geodesic, γ, there is a portion of
the cusp region surrounding each end which is disjoint from all other geodesics
in Gcusp. We call such a portion a gap. The geometry of the gaps attached
to γ is described and their area computed by elementary trigonometry. The
area is a function of the length of the unique closed simple geodesic disjoint
from γ. Next we consider the a generic geodesic in Gcusp, that is, a geodesic
with a single end up the cusp and another end spiralling to a minimal compact
lamination which is not a closed geodesic. We show that such a geodesic is the
limit from both the right and left of other geodesics in Gcusp.
Finally we give
a technique for approximating a geodesic with a single end up the cusp and
the other end spiralling to a closed geodesic. Essentially we repeatedly Dehn
twist a suitable geodesic in Gcusp round this closed geodesic. The results of
this chapter are then combined with a theorem of J. Birman and C. Series to
yield the identity.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Geodesics (Mathematics), Torus (Geometry), Geometry, Hyperbolic, Cusp forms (Mathematics)
Official Date:	May 1991
Dates:	Date Event May 1991 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Epstein, D. B. A.
Sponsors:	Science and Engineering Research Council (Great Britain) (SERC)
Extent:	v, 29 p.
Language:	eng

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