438 research outputs found
A survey of some arithmetic applications of ergodic theory in negative curvature
This paper is a survey of some arithmetic applications of techniques in the
geometry and ergodic theory of negatively curved Riemannian manifolds, focusing
on the joint works of the authors. We describe Diophantine approximation
results of real numbers by quadratic irrational ones, and we discuss various
results on the equidistribution in , and in the
Heisenberg groups of arithmetically defined points. We explain how these
results are consequences of equidistribution and counting properties of common
perpendiculars between locally convex subsets in negatively curved orbifolds,
proven using dynamical and ergodic properties of their geodesic flows. This
exposition is based on lectures at the conference "Chaire Jean Morlet:
G\'eom\'etrie et syst\`emes dynamiques", at the CIRM, Luminy, 2014. We thank B.
Hasselblatt for his strong encouragements to write this survey.Comment: 31 pages, 15 figure
Weakly commensurable groups, with applications to differential geometry
The article contains a survey of our results on weakly commensurable
arithmetic and general Zariski-dense subgroups, length-commensurable and
isospectral locally symmetric spaces and of related problems in the theory of
semi-simple agebraic groups. We have included a discussion of very recent
results and conjectures on absolutely almost simple algebraic groups having the
same maximal tori and finite-dimensional division algebras having the same
maximal subfields.Comment: Improved exposition, updated bibliography. arXiv admin note:
substantial text overlap with arXiv:1212.121
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