Geometry in Groups

If G is a group, its outer-automorphism group Out(G) is obtained from Aut(G) by quotienting out inner automorphisms, that are conjugations by elements of G. It is natural to ask methods and invariants to discuss whether two elements of Out(G) are conjugate. Important examples are GLn(Z) as automorphism group of Zn, Mapping Class Groups as outer-automorphism groups of surface groups, and outer-automorphism groups of finitely generated free groups. The case of the free group of rank 2: Out(F2) is isomorphic to GL(2, Z) and the classification of its conjugacy classes is classical. In rank 3, it is well known that interesting features appear, and they illustrate the rich theory of train tracks, laminations, geometry of suspensions, and structure of the polynomially subgroups, associated to an automorphism. With Francaviglia, Martino, and Touikan, we produced a solution to the conjugacy problem in Out(F3), which is the aim of this mini-course.

References:

‘Introduction to group the...

I will give the basics on property(T), and explain the characterization in terms of actions on median spaces. I will discuss CAT(0) cubical complexes as examples of median spaces, and discuss groups acting on those objects as having a strong negation of property(T).
Possible problems for 2nd week:

Read ‘Spectral interpretations of property(T)’ by Yann Ollivier, with a generalization in mind. http://www.yann-ollivier.org/rech/publs/aut_spec_T.pdf
Study the orbits of the action of discrete cocompact subgroup P in SL^(2,R) on a median space (viewing P as a subgroup of the mapping class group of a surface prevents a proper action on a CAT(0) space). This will involve reading ‘Geometries of 3-manifolds’ by Peter Scott https://deepblue.lib.umich.edu/bitstream/handle/2027.42/135276/blms0401....

Pre-requisites: Bridson-Haefliger Part I, Part II Sections 1, 2 (ideally also 6,8,10), Part III H

If G is a group, its outer-automorphism group Out(G) is obtained from Aut(G) by quotienting out inner automorphisms, that are conjugations by elements of G. It is natural to ask methods and invariants to discuss whether two elements of Out(G) are conjugate. Important examples are GLn(Z) as automorphism group of Zn, Mapping Class Groups as outer-automorphism groups of surface groups, and outer-automorphism groups of finitely generated free groups. The case of the free group of rank 2: Out(F2) is isomorphic to GL(2, Z) and the classification of its conjugacy classes is classical. In rank 3, it is well known that interesting features appear, and they illustrate the rich theory of train tracks, laminations, geometry of suspensions, and structure of the polynomially subgroups, associated to an automorphism. With Francaviglia, Martino, and Touikan, we produced a solution to the conjugacy problem in Out(F3), which is the aim of this mini-course.

References:

‘Introduction to group the...

In 1993 Brink and Howlett proved that the Davis-Shapiro (regular) language provides an automatic structure for Coxeter groups. This means that appropriate paths in the Cayley graph fellow travel, and allows to effectively solve the Word Problem. Similarly, having a bi-automatic structure allows to solve the Conjugacy Problem. However, the Davis-Shapiro language fails to be bi-automatic, even though the Conjugacy Problem for Coxeter groups has been solved by Krammer.

Other languages have been studied over the years, but only recently we came across one (that we call ‘voracious’) giving the bi-automaticity of Coxeter groups. In this minicourse we will explain the proof of our theorem. It involves the Parallel Wall Theorem of Brink and Howlett, the CAT(0) geometry of the Davis complex, and the bipodality of Dyer and Hohlweg.

Pre-requisites: Before the minicourse, please read Sections 1.1, 2.1, and 2.2 of the book ‘Lectures on buildings’ by Ronan.

In geometric group theory, a group is studied via its action on some metric spaces, often with unbounded orbit. It is then natural to investigate the growth of orbit points, group elements, and conjugacy classes. Furthermore, the pioneering works of Patterson and Sullivan and subsequent development connected this growth problem with the dynamics on the boundary and the geodesic flow on the space. In this mini-course, we will study the growth problem in hyperbolic groups and CAT(0) groups via a combinatorial approach. The flexible nature of our approach allows many generalizations.

In week 2, we will compute the growth rate of certain subsets/subgroups of the ambient group. For example, refer to the following articles:
 1. I. Gekhtman and W. Yang, Counting conjugacy classes in groups with contracting elements. (2022)
 2. V. Erlandsson and J. Souto, Counting geodesics of given commutator length. (2023)

Basic references:

Introduction to hyperbolic groups, by Davide Spriano h...

We will study groups acting geometrically on infinite polyhedral complexes. A good example to think of is the fundamental group of a surface Sg acting on the universal cover of a triangulated Sg. We will define, discuss, and state some properties of two very useful invariants of these spaces/actions: the ℓ2-homology of the space and the (Gromov) boundary of the space when it is hyperbolic. There will be many examples, which will hopefully lead to some intuition about these invariants. Possible second week problem based on Matt Clay’s paper ‘ℓ2-homology of the free group’.

Pre-requisites: Background that might be helpful includes some topology and covering space theory (for example Hatcher chapter 1), familiarity with simplicial homology, CW complexes, geometry of H2.

If G is a group, its outer-automorphism group Out(G) is obtained from Aut(G) by quotienting out inner automorphisms, that are conjugations by elements of G. It is natural to ask methods and invariants to discuss whether two elements of Out(G) are conjugate. Important examples are GLn(Z) as automorphism group of Zn, Mapping Class Groups as outer-automorphism groups of surface groups, and outer-automorphism groups of finitely generated free groups. The case of the free group of rank 2: Out(F2) is isomorphic to GL(2, Z) and the classification of its conjugacy classes is classical. In rank 3, it is well known that interesting features appear, and they illustrate the rich theory of train tracks, laminations, geometry of suspensions, and structure of the polynomially subgroups, associated to an automorphism. With Francaviglia, Martino, and Touikan, we produced a solution to the conjugacy problem in Out(F3), which is the aim of this mini-course.

References:

‘Introduction to group the...

We will study groups acting geometrically on infinite polyhedral complexes. A good example to think of is the fundamental group of a surface Sg acting on the universal cover of a triangulated Sg. We will define, discuss, and state some properties of two very useful invariants of these spaces/actions: the ℓ2-homology of the space and the (Gromov) boundary of the space when it is hyperbolic. There will be many examples, which will hopefully lead to some intuition about these invariants. Possible second week problem based on Matt Clay’s paper ‘ℓ2-homology of the free group’.

Pre-requisites: Background that might be helpful includes some topology and covering space theory (for example Hatcher chapter 1), familiarity with simplicial homology, CW complexes, geometry of H2.

I will give the basics on property(T), and explain the characterization in terms of actions on median spaces. I will discuss CAT(0) cubical complexes as examples of median spaces, and discuss groups acting on those objects as having a strong negation of property(T).
Possible problems for 2nd week:

Read ‘Spectral interpretations of property(T)’ by Yann Ollivier, with a generalization in mind. http://www.yann-ollivier.org/rech/publs/aut_spec_T.pdf
Study the orbits of the action of discrete cocompact subgroup P in SL^(2,R) on a median space (viewing P as a subgroup of the mapping class group of a surface prevents a proper action on a CAT(0) space). This will involve reading ‘Geometries of 3-manifolds’ by Peter Scott https://deepblue.lib.umich.edu/bitstream/handle/2027.42/135276/blms0401....

Pre-requisites: Bridson-Haefliger Part I, Part II Sections 1, 2 (ideally also 6,8,10), Part III H