Tag - Geometric group theory
This is a 21-lecture course, with each lecture being either one or two hours, given by Giulio Tiozzo. It gives an introduction to random walks on groups. This class will focus on properties of group actions from a probabilistic point of view, investigating the relations between the dynamics, measure theory and geometry of groups.
We will start with a brief introduction to ergodic theory, discussing measurable transformations and the basic ergodic theorems. Then we will approach random walks on matrix groups and lattices in Lie groups, following the work of Furstenberg. Topics of discussion will be: positivity of drift and Lyapunov exponents. Stationary measures. Geodesic tracking. Entropy of random walks. The Poisson-Furstenberg boundary. Applications to rigidity. We will then turn to a similar study of group actions which do not arise from homogeneous spaces, but which display some features of negatively curved spaces: for instance, hyperbolic groups (in the sense of Gromov) and groups acting on hyperbolic spaces. This will lead us to applications to geometric topology: in particular, to the study of mapping class groups and Out(FN).
Prerequisites: An introduction to measure theory and/or probability, basic topology and basic group theory. No previous knowledge of geometric group theory or Teichmüller theory is needed.
Right-angled Artin groups are perhaps the most ubiquitous manifestations of polyhedral products in geometric group theory and low-dimensional topology. The theory of their subgroups has been of great importance in the last couple of decades. This is especially true with regards to what are known as 'finiteness properties' - meaningful criteria for measuring ways in which infinite groups may behave like finite ones - as well as the theory of three-dimensional manifolds. We will visit some celebrated theorems and, if time allows, discuss problems arising from deck transformations of branched covering maps.
In the 1960s Higman was able to characterize the finitely generated subgroups of finitely presented groups, that is, groups defined using a finite set of generators and finite set of defining relations. His result, which is called the Higman Embedding Theorem, is a key result in combinatorial group theory which makes precise the connection between group presentations and logic. In this talk I will present a result of a similar flavour, proved in recent joint work with Mark Kambites (Manchester), in which we characterise the groups of units of inverse monoids defined by presentation where all the defining relators are of the form w=1. I will explain what an inverse monoid is, the motivation for studying this class of inverse monoids, and also outline some of the geometric ideas that we developed in order to prove our results.
Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner’s theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood. We consider ℤ-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. Surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. In particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of non-trivial horocycle orbit closures is known. Based on joint work with James Farre and Yair Minsky.
This is a 24-lecture course, with each lecture being 75 minutes, given by Slawomir Solecki. Note that the 2nd lecture was not recorded. The other lectures might still be of significant interest, but this needs to be known.
This course focuses on the interaction between set theory, geometry, group theory, and dynamics. It will present parts of Rosendal’s Coarse Geometry of Topological Groups, Kechris-Pestov-Todorcevic’s Fraïssé Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups, as well as theory of Borel and measurable combinatorics.
Given a string Coxeter system (W,S), we construct highly regular quotients of the 1-skeleton of its universal polytope P, which form an infinite family of expander graphs when (W,S) is indefinite and P has finite vertex links. The regularity of the graphs in this family depends on the Coxeter diagram of (W,S). The expansion stems from superapproximation applied to (W,S). This construction is also extended to cover Wythoffian polytopes. As a direct application, we obtain several notable families of expander graphs with high levels of regularity, answering in particular a question posed by Chapman, Linial and Peled positively.
This talk is based on joint work with Marston Conder, Alexander Lubotzky and Francois Thilmany.
This video was produced by the Sydney Mathematical Research Institute, as part of their SMRI seminar series.
Discrete 2-generator subgroups of PSL2(ℝ) have been extensively studied by investigating their action by Möbius transformations on the hyperbolic plane. Due to work of Gilman, Rosenberger, Purzitsky and many others, there is a complete classification of such groups by isomorphism type, and an algorithm to decide whether or not a 2-generator subgroup of PSL2(ℝ) is discrete.
Here we completely classify discrete 2-generator subgroups of PSL2(ℚp) over the p-adic numbers ℚp by studying their action by isometries on the corresponding Bruhat-Tits tree. We give an algorithm to decide whether or not a 2-generator subgroup of PSL2(ℚp) is discrete, and discuss how this can be used to decide whether or not a 2-generator subgroup of SL2(ℚp) is dense.
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