 

Mark Sheingorn
Geodesics on Riemann Surfaces with Ramification Points of Order Greater than Two


Published: 
October 17, 2001

Keywords: 
geodesy, ramification, elliptic fixed point 
Subject: 
30F10,35,45 11F06 


Abstract
Most commonly, studies of geodesy on Riemann surfaces
proceed on those surfaces without ramification. While it is true that
every surface of finite type (stemming from a finitely generated
Fuchsian group) has a finitesheeted cover which is free of
ramification, local geometric information is lost in this process.
We seek to analyze geodesics on ramified surfaces
directly. After briefly reviewing (the by now wellunderstood) situation
of ramification points of order 2, we turn to higher ramification.
Examples are offered on the surface stemming from the squares of
elements of the full modular group Γ^{2}\H of
signature (0;3,3,∞) . Already here we can see that a fundamental
property of negatively curved surfaces fails: there are closed curves
with apparently nontrivial homotopy, yet having no geodesic in their
homotopy class. Next, we construct surfaces on which length L and
number of selfintersections N of a closed bounded height geodesic are
closely linked: There are constants a and b depending only the
surface and said height such that aL(τ) < bN(τ) < L(τ). On
such surfaces, long simple geodesic arcs cannot exist.


Acknowledgements
Research supported by Princeton University and PSCCUNY


Author information
CUNY  Baruch College, New York, NY 10010
marksh@alum.dartmouth.org
http://www.panix.com/~marksh/

