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Monthly Archives: April 2017
Notes of Pierre-Emmanuel Caprace’s fourth Cambridge lecture 27-04-2017
Exotic lattices and simple locally compact groups, IV 1. Subgroup separability and residual finiteness Theorem 1 (Kropholler-Reid-Wesolek-Caprace) Let be a finitely generated group, let be a commensurated subgroup (every conjugate commensurates ). Let be the profinite closure of , i.e. … Continue reading
Notes of Emmanuel Breuillard’s sixth Cambridge lecture 26-04-2017
Introduction to approximate groups, VI Recall the weak version of the structure theorem for approximate groups. Theorem 1 (Breuillard-Green-Tao) Let . Let be a finite group such that . Then there exists a virtually nilpotent subgroup of and such that … Continue reading
Notes of Marc Burger’s fourth Leverhulme lecture 25-04-2017
Towards higher Teichmuller theory, IV 1. To infinity and beyond Today, . We consider maximal representations . They are quasi-isometric embeddings. The class is deformation invariant. If , they coincide with holonomies of hyperbolic structures on . The goal is … Continue reading
Notes of John Parker’s Cambridge lecture 25-04-2017
Non arithmetic lattices With Martin Deraux and Julien Paupert. 1. Arithmetic lattices We are interested in lattices in . One source of examples is the following arithmetic construction due to Borel-Harish Chandra. Let be an algebraic linear group defined over … Continue reading
Notes of Frederic Paulin’s Cambridge lecture 25-04-2017
On continued fractions, diagonal group actions and trees in positive characteristic With Uri Shapira. 1. Continued fractions Consider the ring of 1-variable polynomials over a finite field. Its fraction field can be completed with respect to the absolute value into … Continue reading
Notes of Marc Burger’s third Leverhulme lecture 24-04-2017
Towards higher Teichmuller theory, III 1. On integer points in the Hitchin moduli space Let be a simple Lie group of real split type (…). Then has a Hitchin component. In the sequel, we focus on . The Hitchin component … Continue reading
Notes of Brian Bowditch’s Cambridge lecture 20-04-2017
Bounding genera of singular surfaces Ultimate motivation: understand the curve complex for non-compact surfaces. But today, only closed surfaces around. 1. Genus distance The curve complex of is hyperbolic, pseudo-Anosov diffeos act loxodromically, moving vertices at a linear speed. Say … Continue reading
Notes of Andreas Aaserud’s Cambridge lecture 20-04-2017
Property (T) and approximate conjugacy of actions Joint work with Sorin Popa. We let countable groups act on probability spaces. We focus on free and ergodic actions. Example. Bernoulli action on . 1. Actions on von Neumann algebras Von Neumann … Continue reading
Notes of Emmanuel Breuillard’s fifth Cambridge lecture 19-04-2017
Introduction to approximate groups, V Today, I explain mathematics from the 1950’s: the solution of Hilbert’s 5th problem. Indeed, ideas from this old theory enters our structure theory of approximate groups. 1. Hilbert’s 5th problem In 1900, Hilbert wondered wether … Continue reading
Notes of Mikolai Fraczyk’s Cambridge lecture 19-04-2017
Benjamin-Schramm convergence of arithmetic 3-manifolds 1. B-S convergence and limit multiplicity Let be a semi-simple Lie group, let be a sequence of lattices in , without infinite repetitions. We say that locally symmetric spaces B-S converge if the relative volume … Continue reading
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