117 research outputs found
Deformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers
A connection between real poles of the growth functions for Coxeter groups
acting on hyperbolic space of dimensions three and greater and algebraic
integers is investigated. In particular, a geometric convergence of fundamental
domains for cocompact hyperbolic Coxeter groups with finite-volume limiting
polyhedron provides a relation between Salem numbers and Pisot numbers. Several
examples conclude this work.Comment: 26 pages, 16 figures, 4 data tables; minor corrections; European
Journal of Combinatorics, 201
Volume of a doubly truncated hyperbolic tetrahedron
The present paper regards the volume function of a doubly truncated
hyperbolic tetrahedron. Starting from the previous results of J. Murakami, U.
Yano and A. Ushijima, we have developed a unified approach to express the
volume in different geometric cases via dilogarithm functions and to treat
properly the many analytic strata of the latter. Finally, several numeric
examples are given.Comment: Several misprints in the proof of Theorem 1 correcte
Free subgroups of free products and combinatorial hypermaps
We derive a generating series for the number of free subgroups of finite
index in by using a connection between
free subgroups of and certain hypermaps (also known as ribbon graphs
or "fat" graphs), and show that this generating series is transcendental. We
provide non-linear recurrence relations for the above numbers based on
differential equations that are part of the Riccati hierarchy. We also study
the generating series for conjugacy classes of free subgroups of finite index
in , which correspond to isomorphism classes of hypermaps. Asymptotic
formulas are provided for the numbers of free subgroups of given finite index,
conjugacy classes of such subgroups, or, equivalently, various types of
hypermaps and their isomorphism classes.Comment: 27 pages, 3 figures; supplementary SAGE worksheets available at
http://sashakolpakov.wordpress.com/list-of-papers
Counting cusped hyperbolic 3-manifolds that bound geometrically
We show that the number of isometry classes of cusped hyperbolic
-manifolds that bound geometrically grows at least super-exponentially with
their volume, both in the arithmetic and non-arithmetic settings.Comment: 17 pages, 7 figures; to appear in Transactions AM
Hyperbolic four-manifolds, colourings and mutations
We develop a way of seeing a complete orientable hyperbolic -manifold
as an orbifold cover of a Coxeter polytope that has a facet colouring. We also develop a way of finding
totally geodesic sub-manifolds in , and describing
the result of mutations along . As an application of our method,
we construct an example of a complete orientable hyperbolic -manifold
with a single non-toric cusp and a complete orientable hyperbolic
-manifold with a single toric cusp. Both and
have twice the minimal volume among all complete orientable
hyperbolic -manifolds.Comment: 24 pages, 11 figures; to appear in Proceedings of the London
Mathematical Societ
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