1,603 research outputs found
Equivelar and d-Covered Triangulations of Surfaces. I
We survey basic properties and bounds for -equivelar and -covered
triangulations of closed surfaces. Included in the survey is a list of the
known sources for -equivelar and -covered triangulations. We identify all
orientable and non-orientable surfaces of Euler characteristic
which admit non-neighborly -equivelar triangulations
with equality in the upper bound
. These
examples give rise to -covered triangulations with equality in the upper
bound . A
generalization of Ringel's cyclic series of neighborly
orientable triangulations to a two-parameter family of cyclic orientable
triangulations , , , is the main result of this
paper. In particular, the two infinite subseries and
, , provide non-neighborly examples with equality for
the upper bound for as well as derived examples with equality for the upper
bound for .Comment: 21 pages, 4 figure
The Moduli space of Riemann Surfaces of Large Genus
Let be the -thick part of the moduli
space of closed genus surfaces. In this article, we show
that the number of balls of radius needed to cover
is bounded below by and bounded above
by , where the constants depend only on and
, and in particular not on . Using the counting result we prove that
there are Riemann surfaces of arbitrarily large injectivity radius that are not
close (in the Teichm\"uller metric) to a finite cover of a fixed closed Riemann
surface. This result illustrates the sharpness of the Ehrenpreis conjecture.Comment: v2, accepted in GAFA, updates based on referee's comment
Some Triangulated Surfaces without Balanced Splitting
Let G be the graph of a triangulated surface of genus . A
cycle of G is splitting if it cuts into two components, neither of
which is homeomorphic to a disk. A splitting cycle has type k if the
corresponding components have genera k and g-k. It was conjectured that G
contains a splitting cycle (Barnette '1982). We confirm this conjecture for an
infinite family of triangulations by complete graphs but give counter-examples
to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should
contain splitting cycles of every possible type.Comment: 15 pages, 7 figure
The Gromov Norm of the Product of Two Surfaces
We make an estimation of the value of the Gromov norm of the Cartesian
product of two surfaces. Our method uses a connection between these norms and
the minimal size of triangulations of the products of two polygons. This allows
us to prove that the Gromov norm of this product is between 32 and 52 when both
factors have genus 2. The case of arbitrary genera is easy to deduce form this
one.Comment: The journal version contains an error that invalidates one direction
of the main theorem. The present version contains an erratum, at the end,
explaining thi
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