Publications
- A lower bound for
the volume of hyperbolic 4-orbifolds
In progress.
- On volumes of
complex hyperbolic orbifolds
(with Guofang Wei), in
preparation.
- On volumes of hyperbolic orbifolds
arXiv:0911.4712v1
[math.GT]
(with Guofang Wei),
submitted.
Abstract: Denote by G the Lie group of orientation-preserving isometries of
hyperbolic
n-space endowed
with the canonical Riemannian metric. Let Γ
be
a
discrete subgroup
of
G. There exists a Riemannian submersion π from G into hyperbolic n-space. It follows
that the covolume of Γ in hyperbolic space is the covolume
of Γ in G
divided by volume
of the special orthogonal group, which is the preimage of any point under π.
By a theorem due to H. C. Wang, valid for any semisimple Lie group
without
compact factor, the
covolume of Γ in G is bounded below by the volume
of metric ball whose radius is dependent
only on the root system of the Lie algebra of G.
In this
paper, we derive the curvature formulas
for the canonical metric of a semisimple Lie group. We use these
formulas to construct an
upper bound for the sectional curvature of G. This result, together
with Gunther's comparison
theorem, produces an explicit lower bound for a hyperbolic orbifold
dependent on dimension
alone.
- Lower bounds for the volume of hyperbolic n-orbifolds
arXiv:0709.0311v1
[math.GT]
Pacific
Journal
of
Mathematics 237 (2008), no. 1, 1--19.
Abstract: Let Γ
be a nonelementary group of
orientation-preserving isometries of hyperbolic
n-space. Extensions of Jørgensen's
inequality to all dimensions, due to Martin and Friedland
& Hersonsky, provided necessary
conditions for the discreteness of Γ. These conditions were
subsequently used to construct a lower bound for the radius of an
embedded ball in the
quotient space when Γ is assumed to be torsion-free.
In this paper, the torsion-free assumption is replaced with a upper
bound, k, on the maximal
order of torsion. An analysis based on the hyperbolic distance between
a marked point in
hyperbolic space and the fixed point set of an elliptic isometry
establishes a uniform normed
distance from the identity for the elements of an orbifold fundamental
group. A combinatorial
argument is then used to estimate the degree of the cover of an
embedded ball. The result is
an explicit formula for the volume of a hyperbolic orbifold dependent on n and k.