460 research outputs found
Arithmetic lattices and weak spectral geometry
This note is an expansion of three lectures given at the workshop "Topology,
Complex Analysis and Arithmetic of Hyperbolic Spaces" held at Kyoto University
in December of 2006 and will appear in the proceedings for this workshop.Comment: To appear in workshop proceedings for "Topology, Complex Analysis and
Arithmetic of Hyperbolic Spaces". Comments welcom
Cusps of arithmetic orbifolds
This thesis investigates cusp cross-sections of arithmetic real, complex, and
quaternionic hyperbolic --orbifolds. We give a smooth classification of
these submanifolds and analyze their induced geometry. One of the primary tools
is a new subgroup separability result for general arithmetic lattices.Comment: 76 pages; Ph.D. thesi
Asymptotic growth and least common multiples in groups
In this article we relate word and subgroup growth to certain functions that
arise in the quantification of residual finiteness. One consequence of this
endeavor is a pair of results that equate the nilpotency of a finitely
generated group with the asymptotic behavior of these functions. The second
half of this article investigates the asymptotic behavior of two of these
functions. Our main result in this arena resolves a question of Bogopolski from
the Kourovka notebook concerning lower bounds of one of these functions for
nonabelian free groups.Comment: 13 page
Primitive geodesic lengths and (almost) arithmetic progressions
In this article, we investigate when the set of primitive geodesic lengths on
a Riemannian manifold have arbitrarily long arithmetic progressions. We prove
that in the space of negatively curved metrics, a metric having such arithmetic
progressions is quite rare. We introduce almost arithmetic progressions, a
coarsification of arithmetic progressions, and prove that every negatively
curved, closed Riemannian manifold has arbitrarily long almost arithmetic
progressions in its primitive length spectrum. Concerning genuine arithmetic
progressions, we prove that every non-compact, locally symmetric, arithmetic
manifold has arbitrarily long arithmetic progressions in its primitive length
spectrum. We end with a conjectural characterization of arithmeticity in terms
of arithmetic progressions in the primitive length spectrum. We also suggest an
approach to a well known spectral rigidity problem based on the scarcity of
manifolds with arithmetic progressions.Comment: v3: 23 pages. To appear in Publ. Ma
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