An LMS online lecture course in mapping class groups.
Tag - Geometric group theory
An LMS online lecture course in Kazhdan's property (T).
The study of groups with "hyperbolic-like directions" has been a central theme in geometric group theory. Two notions are usually used to quantify what is meant by "hyperbolic-like directions", the notion of a contracting geodesic and that of a Morse geodesic. Since the property that every geodesic ray in metric space is contracting or Morse characterizes hyperbolic spaces, being a contracting/Morse geodesic is considered a hyperbolic-like property. Generalizing work of Cannon, I will discuss a joint result with Eike proving that for any finitely generated group, the language of contracting geodesics with a fixed parameter is a regular language. This immediately implies that contracting geodesics can't exist in torsion groups.
The Morse notion is a weaker notion than that of the contracting notion, in fact, building on work of Osin, Ol’shanskii, and Sapir, Fink gave an example of a torsion group which contains an infinite Morse geodesic. This seems to contradict the claim that Morse geodesics are "hyperbolic-like" directions. As an attempt to rectify this, Russell, Spriano, and Tran introduced a class of spaces where Morse (quasi)-geodesics satisfy some local-to-global property and they showed that many interesting examples live in such a class. In these spaces, Morse (quasi)-geodesics are expected to behave more reasonably like "hyperbolic directions", therefore, such spaces/groups can be regarded as good hosts of Morse (quasi)-geodesics.
I will discuss some continuation of their work where we show that in such spaces Morse geodesics form a regular language, give a characterization of stable subgroups in terms of regular languages. Time permitting, I will discuss few other applications of these automatic structures to the growth of stable subgroups and the dynamics of the action of such groups on their Morse boundaries. This work is joint with Cordes, Russell and Spriano.
A finite graph that can be obtained from a given graph by contracting edges and removing vertices and edges is said to be a minor of this graph. Minors have played an important role in graph theory, ever since the well-known result of Kuratowski that characterized planar graphs as those that do not admit the complete graph on five vertices nor the complete bipartite graph on (3,3) vertices as minors. In this talk, we will explore how this concept interacts with some notions from geometric group theory, and describe a new characterisation of virtually free groups in terms of minors of their Cayley graphs.
We show that for almost all primitive integral cohomology classes in the fibred cone of a closed fibred hyperbolic 3-manifold, the monodromy normally generates the mapping class group of the fibre. The key idea of the proof is to use Fried’s theory of suspension flow and dynamic blow-up of Mosher. If the time permits, we also discuss the non-existence of the analogue of Fried’s continuous extension of the normalized entropy over the fibered face in the case of asymptotic translation lengths on the curve complex.
We explain the proof that Neretin groups have no non-trivial ergodic invariant random subgroups (IRS). Equivalently, any non-trivial ergodic p.m.p. action of Neretin’s group is essentially free. This property can be thought of as simplicity in the sense of measurable dynamics; while Neretin groups were known to be abstractly simple by a result of Kapoudjian. The heart of the proof is a 'double commutator' lemma for IRSs of elliptic subgroups.
The notion of a "hierarchically hyperbolic space/group" grows out of geometric similarities between CAT(0) cubical groups and mapping class groups. Hierarchical hyperbolicity is a "coarse nonpositive curvature" property that is more restrictive than acylindrical hyperbolicity but general enough to include many of the usual suspects in geometric group theory. The class of hierarchically hyperbolic groups is also closed under various procedures for constructing new groups from old, and the theory can be used, for example, to bound the asymptotic dimension and to study quasi-isometric rigidity for various groups. One disadvantage of the theory is that the definition - which is coarse-geometric and just an abstraction of properties of mapping class groups and cube complexes - is complicated. We therefore present a comparatively simple sufficient condition for a group to be hierarchically hyperbolic, in terms of an action on a hyperbolic simplicial complex. I will discuss some applications of this criterion to mapping class groups and (non-right-angled) Artin groups.
Hierarchically hyperbolic groups (HHGs) and spaces are recently introduced generalizations of (Gromov-) hyperbolic groups and spaces. Other examples of HHGs include mapping class groups, right-angled Artin/Coxeter groups, and many groups acting properly and cocompactly on CAT(0) cube complexes. After a substantial introduction and motivation, I will present a combination theorem for hierarchically hyperbolic groups. As a corollary, any graph product of finitely many HHGs is itself a HHG.
Bridson, Burillo, Elder and Šunić asked if there exists a group with intermediate geodesic growth and if there is a characterization of groups with polynomial geodesic growth. Towards these questions, they showed that there is no nilpotent group with intermediate geodesic growth, and they provided a sufficient condition for a virtually abelian group to have polynomial geodesic growth. In this talk, we take the next step in this study and show that the geodesic growth for a finitely generated virtually abelian group is either polynomial or exponential; and that the generating function of this geodesic growth series is holonomic, and rational in the polynomial growth case. To obtain this result, we will make use of the combinatorial properties of the class of linearly constrained language as studied by Massazza. In addition, we show that the language of geodesics of a virtually abelian group is blind multicounter.
We will survey new and old results on the "energy'' of a set of affine transformations and see applications in the geometric side of additive combinatorics, in combinatorial geometry and in questions on growth in the affine group. In particular we will answer to the affirmative a question of Yufei Zhao.
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