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Monthly Archives: March 2017
Notes of Alan Reid’s third Oxford lecture 24-03-2017
Profinite rigidity in low dimensions, III 1. Profinite rigidity of classes of closed 3-manifolds We are interested in the restricted genus Theorem 1 (Wilton-Zalesskii) Let , be the fundamental groups of closed 3-manifolds, with . If one is hyperbolic, so … Continue reading
Notes of Enrico Le Donne’s Oxford lecture 24-03-2017
From homogeneous spaces to Lie groups With Michael Cowling, Ville Kivioja, Alessandro Ottazi and Sebastiano Nicolussi Golo. 1. Homogeneous metric spaces I.e. with a transitive group of isometries. Solenoids (inverse limits of with maps ) are homogeneous. -adics are too. … Continue reading
Notes of Emmanuel Breuillard’s third Oxford lecture 24-03-2017
Approximate groups, III 1. Growth Let be a finitely generated free group. It is easy that for the standard generating system, . What about other generating systems? Theorem 1 (Razborov, Safin) For an arbitrary generating set in , If … Continue reading
Notes of Bruno Duchesne’s Oxford lecture 23-03-2017
Groups acting on dendrites Joint with Nicolas Monod. 1. Dendrites A dendrite is a continuum (compact metrizable connected spaces) which is locally connected and in which every two points are joined by a unique arc (continuous injective image of an … Continue reading
Notes of Marc Burger’s third Oxford lecture 23-03-2017
Geometric structures, compactifications of representation varieties, and non archimedean geometry, III 1. To infinity (and beyond) 1.1. Compactification We intend to describe the boundary of the representation variety . Theorem 1 (Burger-Pozzetti) If is a maximal representation, then Nontrivial elements … Continue reading
Notes of Alireza Salehi Golsefidy’s Oxford lecture 23-03-2017
Super-approximation 1. Definition Say a finitely generated subgroup of (with fixed generating set ) has super-approximation with respect to a set of positive integers which are coprime to if the family of Cayley graphs is an expander. Here, Example. When … Continue reading
Notes of Alan Reid’s second Oxford lecture 23-03-2017
Profinite rigidity in low dimensions, II 1. Profinite completion We organize finite quotients of a group . The set of normal finite index subgroups of is a directed set, where if contains (unfortunate conflict of notation). The profinite completion of … Continue reading
Notes of Jon Chaika’s Oxford lecture 22-03-2017
Joinings of dynamical systems We deal with probability measure preserving maps. Measures control points. For instance, Birkhoff’s theorem states that time averages of a function converge a.e. In the uniquely ergodic case, convergence holds everywher for continuous functions. 1. Joinings … Continue reading
Notes of Marc Burger’s second Oxford lecture 22-03-2017
Geometric structures, compactifications of representation varieties, and non archimedean geometry, II 1. Maximality 1.1. Toledo invariants On , preserves a Riemannian metric and a complex structure , hence a non-degenerate differential 2-form . It turns out that . Indeed, should … Continue reading
Notes of Goulnara Arzhantseva’s third Oxford lecture 22-03-2017
Constructions of non -exact groups, III Although we seem to construct counterexamples, observe that the technique I explained is used for positive results. 1. Non -exact groups with Haagerup property Amenability implies Coarse amenability which implies Coarse embeddability. In turn, … Continue reading
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