This is website for the geometric group theory seminar at IASM of Harbin Institute of Technology. There will be weekly event consisting of either seminar talks or mini-courses. These will all be streamed via zoom, recorded and uploaded here afterwards. This is the Zoom link for the seminar, and the passcode is : 6 7 9 0 9 1, no space in between.

While most will happen in person, some of the talks will be virtual, so if you are interested in giving a zoom talk at anytime, feel free to email me at: zalloumabdul at gmail dot com.

Contact: **ABDALRAZZAQ R.A. ZALLOUM** （zalloumabdul@gmail.com）

**Upcoming talks**

**Friday, November 8th, 10 am (Chinese time)**

**Speaker:** Lihuang Ding(Peking University)

**Title: **Growth Tightness of Groups with Contracting Elements and Confined Subgroups

## Abstract

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.

**Mini Course, November 13th-15th**

**Speaker:** Wenyuan Yang(Peking University)

**Title: **Growth of finitely generated groups

## Abstract

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.

**Friday, November 22th, 10 am (Chinese time)**

**Speaker:** Robert Tang(Xi'an Jiaotong-Liverpool University)

**Title: **TBA

## Abstract

TBA

**Friday, November 29th, 10 am (Chinese time)**

**Speaker:** TBA

**Title: **TBA

## Abstract

TBA

**Mini Course, December 1st-14th**

**Speaker:** Nir Lazarovich(Technion-Israel Institute of Technology)

**Title: **TBA

## Abstract

TBA

**Past talks**

**Tuesday, September 24th, 10-11 am （Chinese Time）and Friday, September 27th, 10-11 am （Chinese time）**

**Speaker: **Renxing Wan (East China Normal University)

**Title:** Marked length spectrum rigidity in geometric group theory

## Abstract，video 1，video 2

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.

**Friday, October 11th, 10 am （Chinese time）**

**Speaker:** Tarik Aougab (Haverford College)

**Title: **Metric entropy of subgraph

## Abstract

## We prove that there is some constant C so that in any finite metric graph with unit entropy, there exists a proper subgraph with entropy at least C, where C depends only on the rank of the original graph. In addition to being independently interesting from a dynamical point of view, this has applications to the study of pressure-type metrics on the Culler-Vogtmann outer space. We will not assume any advanced knowledge in either dynamics or the geometric group theory of Out(Fn). This represents joint work with Tawfiq Ahmed and Matt Clay.

**Friday, October 18th, 4 pm (Chinese time)**

**Speaker:** Macarena Arenas(University of Cambridge)

**Title:**Presenting groups via cubes

## Abstract

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.

**Friday, October 25th, 10 am (Chinese time)**

**Speaker:** Eduardo Reyes (Yale University)

**Title:** Manhattan curves and manifolds and metric structures on hyperbolic groups

## Abstract

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.

**Friday, November 1st, 10 am (Chinese time)**

**Speaker:**Abdul Zalloum (Harbin Institute of Technology)

**Title: **Some aspects of Gromov hyperbolicity

## Abstract

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.