几何群论研讨班

Given a metric space X and a scale parameter σ ≥ 0, the Rips graph RipsσX has X as its vertex set, with two vertices declared adjacent whenever their distance is at most σ. A classical fact is that X is a quasigeodesic space precisely if it is quasi-isometric to its Rips graph at sufficiently large scale. 

By considering all possible scales, we obtain a directed system of graphs known as the Rips filtration. How does the large-scale geometry of RipsσX evolve as σ → ∞? Is there a meaningful notion of limit? It turns out that the answers depends on whether we work up to quasi-isometry or coarse equivalence. In this talk, I will discuss some results and applications inspired by these questions.

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        This talk will be divided into two parts. In the first part, we recall some  basic material in GGT, including marked length spectrum, contracting subsets, admissible path, and Extension Lemma. We also show how to use the Extension Lemma to get a simultaneously contracting element for two group actions with contracting property. 

        In the second part, we prove that if the two actions have the same marked length spectrum, then the orbit map G.x --> G.y must be a rough isometry.  In addition, we prove a finer marked length spectrum rigidity from confined subgroups and further, geometrically dense subgroups. 

        Our proof is based on the Extension Lemma and uses purely elementary metric geometry. This is joint work with Xiaoyu Xu and Wenyuan Yang.

 We’ll explore the problem of finding effective models for the classifying spaces of certain quotients of fundamental groups of non-positively curved cube complexes, we’ll discuss the framework -- cubical small-cancellation theory -- that provides the necessary tools to do so, and, time permitting, we’ll explain how this viewpoint allows us to compute the homology and cohomology of various examples. 

  Hyperbolic groups are generalizations of finitely generated free groups and surface groups, and act properly and cocompactly by isometries on Gromov hyperbolic spaces. For a fixed hyperbolic group, we can pack all its actions of this type into a single space, called space of metric structures, which extends the classically studied Teichmuller and Outer spaces but contains many more interesting actions. I will talk about the geometry of this space when equipped with a natural metric resembling Thurston’s distance on Teichmuller space. The key feature is the notion of Manhattan curves, which allows us to interpolate between two isometric actions and produce plenty of bi-infinite geodesics in the space of metric structures. If time permits, I will talk about a higher-dimensional version of these curves, allowing us to find higher-dimensional manifolds in the space of metric structures. This is joint work with Stephen Cantrell and Cagri Sert.

  I will introduce the concept of a Gromov hyperbolic group and explore its fascinating connections to various fields of mathematics, including geometry, dynamics,regular languages, and automatic structures. This talk is designed for a general audience, and no prior knowledge is required beyond the basic definition of a group.

  Recall that for a hyperbolic group, the quotient group of any infinite normal subgroup has strictly smaller growth rate.Group action with contracting elements is a generalization of a hyperbolic group action. In this talk, we will present a generalization of the growth tightness theorem in group with contracting elements. We will also discuss the growth of confined subgroups. This is joint work with Wenyuan Yang.

In his 1968 paper, Milnor initiated the study of growth of finitely generated groups, which has continued to be an active research topic nowadays in Geometric Group Theory. The growth function of a finitely generated group counts the number of elements in a growing ball in word metric, reminiscent of the classical lattice counting problem in geometry. In this minicourse, we shall focus on groups with exponential growth function, and study what a generic element looks like when groups have certain coarse negative curvature.

The plan of the minicourse is as follows:

Lecture 1: we introduce the notion of Cayley graph and revisit Milnor's argument, now called Milnor-Svarc Lemma, the fundamental lemma of GGT. The growth classification of finitely generated groups will be described.

Lecture 2: we discuss by examples various classes of negatively curved groups, including (relatively) hyperbolic groups and acylindrically hyperbolic groups. A particular emphasis shall be given to the notion of strongly contracting elements and other invariants.

Lecture 3: we will explain how the contracting property could be useful to study counting problems, such as estimating growth function and understanding generic elements.