903 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 complexity of the normal surface solution space
Normal surface theory is a central tool in algorithmic three-dimensional
topology, and the enumeration of vertex normal surfaces is the computational
bottleneck in many important algorithms. However, it is not well understood how
the number of such surfaces grows in relation to the size of the underlying
triangulation. Here we address this problem in both theory and practice. In
theory, we tighten the exponential upper bound substantially; furthermore, we
construct pathological triangulations that prove an exponential bound to be
unavoidable. In practice, we undertake a comprehensive analysis of millions of
triangulations and find that in general the number of vertex normal surfaces is
remarkably small, with strong evidence that our pathological triangulations may
in fact be the worst case scenarios. This analysis is the first of its kind,
and the striking behaviour that we observe has important implications for the
feasibility of topological algorithms in three dimensions.Comment: Extended abstract (i.e., conference-style), 14 pages, 8 figures, 2
tables; v2: added minor clarification
Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds
We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form
of Thurston's Virtual Fibration Conjecture. In particular, this manifold has
finite covers which fiber over the circle in arbitrarily many ways. More
precisely, it has a tower of finite covers where the number of fibered faces of
the Thurston norm ball goes to infinity, in fact faster than any power of the
logarithm of the degree of the cover, and we give a more precise quantitative
lower bound. The example manifold M is arithmetic, and the proof uses detailed
number-theoretic information, at the level of the Hecke eigenvalues, to drive a
geometric argument based on Fried's dynamical characterization of the fibered
faces. The origin of the basic fibration of M over the circle is the modular
elliptic curve E=X_0(49), which admits multiplication by the ring of integers
of Q[sqrt(-7)]. We first base change the holomorphic differential on E to a
cusp form on GL(2) over K=Q[sqrt(-3)], and then transfer over to a quaternion
algebra D/K ramified only at the primes above 7; the fundamental group of M is
a quotient of the principal congruence subgroup of level 7 of the
multiplicative group of a maximal order of D. To analyze the topological
properties of M, we use a new practical method for computing the Thurston norm,
which is of independent interest. We also give a non-compact finite-volume
hyperbolic 3-manifold with the same properties by using a direct topological
argument.Comment: 42 pages, 7 figures; V2: minor improvements, to appear in Amer. J.
Mat
Local limits of uniform triangulations in high genus
We prove a conjecture of Benjamini and Curien stating that the local limits
of uniform random triangulations whose genus is proportional to the number of
faces are the Planar Stochastic Hyperbolic Triangulations (PSHT) defined in
arXiv:1401.3297. The proof relies on a combinatorial argument and the
Goulden--Jackson recurrence relation to obtain tightness, and probabilistic
arguments showing the uniqueness of the limit. As a consequence, we obtain
asymptotics up to subexponential factors on the number of triangulations when
both the size and the genus go to infinity.
As a part of our proof, we also obtain the following result of independent
interest: if a random triangulation of the plane is weakly Markovian in the
sense that the probability to observe a finite triangulation around the
root only depends on the perimeter and volume of , then is a mixture of
PSHT.Comment: 36 pages, 10 figure
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