Pure Maths Colloquium: Asymptotics of Torsion Homology of Hyperbolic 3-Manifolds
6 October 2014 16:00 in CM221
Hyperbolic 3-manifolds have been studied intensely by topologists since
the mid-1970's. When the fundamental group arises from a certain number
theoretic construction (in this case, the manifold is called "arithmetic"), the manifold acquires extra features that lead to important connections with number theory. Accordingly, arithmetic
hyperbolic 3-manifolds have been studied by number theorists (perhaps
not as intensely as the topologists) with different motivations.
Very recently, number theorists have started to study the torsion in the
homology of arithmetic hyperbolic 3-manifolds. The aim of the first half
of this introductory talk, where we will touch upon notions like
"arithmeticity", "Hecke operators", will be to illustrate the importance of torsion from the perspective of number theory. In the second half, I will present new joint work with N.Bergeron and A.Venkatesh which relates the topological complexity of homology cycles to the asymptotic growth of torsion in the homology. I will especially focus on the interesting use of the celebrated "Cheeger-Mueller Theorem" from global analysis.
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