Monday, 29 July 2024

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 theory’ by Oleg Bogopolski
- ‘Classification of automorphisms of the free group of rank 2 by ranks of fixed-point subgroups’ by Oleg Bogopolski
- ‘The Conjugacy Problem for Out(F3)’, by Fran¸cois Dahmani, Stefano Francaviglia, Armando Martino, Nicholas Touikan

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

Tuesday, 30 July 2024

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 https://www.advgrouptheory.com/journal/Volumes/15/Spriano.pdf
- The mini-course will not require knowledge in automatic structure on groups or Patterson-Sullivan theory, though such knowledge will go well with the course. For those who are interested in this direction, I will just recommend Danny Calegari’s note ‘The ergodic theory of hyperbolic groups’: https://arxiv.org/abs/1111.0029v2

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.

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.

Wednesday, 31 July 2024

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 theory’ by Oleg Bogopolski
- ‘Classification of automorphisms of the free group of rank 2 by ranks of fixed-point subgroups’ by Oleg Bogopolski
- ‘The Conjugacy Problem for Out(F3)’, by Fran¸cois Dahmani, Stefano Francaviglia, Armando Martino, Nicholas Touikan

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

Thursday, 01 August 2024

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.

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 theory’ by Oleg Bogopolski
- ‘Classification of automorphisms of the free group of rank 2 by ranks of fixed-point subgroups’ by Oleg Bogopolski
- ‘The Conjugacy Problem for Out(F3)’, by Fran¸cois Dahmani, Stefano Francaviglia, Armando Martino, Nicholas Touikan

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 https://www.advgrouptheory.com/journal/Volumes/15/Spriano.pdf
- The mini-course will not require knowledge in automatic structure on groups or Patterson-Sullivan theory, though such knowledge will go well with the course. For those who are interested in this direction, I will just recommend Danny Calegari’s note ‘The ergodic theory of hyperbolic groups’: https://arxiv.org/abs/1111.0029v2

Friday, 02 August 2024

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

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 https://www.advgrouptheory.com/journal/Volumes/15/Spriano.pdf
- The mini-course will not require knowledge in automatic structure on groups or Patterson-Sullivan theory, though such knowledge will go well with the course. For those who are interested in this direction, I will just recommend Danny Calegari’s note ‘The ergodic theory of hyperbolic groups’: https://arxiv.org/abs/1111.0029v2

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.

Monday, 05 August 2024

By works of Agol, Bergeron–Wise, Dufour, and Kahn–Markovic, among others, it is known that fundamental groups of fibred hyperbolic 3-manifolds (and those of closed hyperbolic 3-manifolds in general) act geometrically on CAT(0) cube complexes and thus these manifolds possess strong group theoretic and topological properties. However, it is unknown if these results extend to the fundamental groups of all hyperbolic surface bundles over finite graphs (in other words, arbitrary hyperbolic surface-by-free groups). In this talk, we will describe how, after modifying these bundles by drilling (that is, removing small tubular neighbourhoods of) enough simple closed curves in specific fibres, the corresponding fundamental groups become hyperbolic relative to the introduced tori subgroups, and as a consequence, act geometrically on CAT(0) cube complexes. This is joint work with Mahan Mj.

Let F be a finite-rank free group with rank greater than 2, Q be a finitely generated subgroup of the outer automorphism group of F, and consider a short exact sequence

1 → F → EQ → Q → 1.

My talk will discuss some known results and some open questions regarding the geometry of EQ, given information about Q. There is a somewhat decent understanding of the geometry of EQ when Q is cyclic, where we have necessary and sufficient conditions on Q to determine if EQ is hyperbolic or relatively hyperbolic or thick. One key idea we will look into is – why EQ is relatively hyperbolic when Q is an infinite cyclic group generated by an exponentially growing outer automorphism of F and then discuss some difficulties one faces while extending such results to the cases when Q is not cyclic.

Tuesday, 06 August 2024

A group G is called acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. It includes many examples of interest, e.g., nonelementary hyperbolic and relatively hyperbolic groups, all but finitely many mapping class groups of punctured closed surfaces, most 3-manifold groups and Out(Fn) for n > 1. Although BS(p, q) is essentially never acylindrically hyperbolic we will see in this talk that Out(BS(p, q)) is acylindrically hyperbolic for non-solvable Baumslag-Solitar groups, and explore further properties of the group using its acylindricity. This is joint work with Daxun Wang.

A random walk on a Fuchsian group (that is, a lattice in SL(2,R)) gives a random walk in the hyperbolic plane. Under mild conditions, a typical sample path for such a walk converges to the circle at infinity. The distribution of the limits of sample paths define a stationary measure on the circle. This talk will survey the landscape of results related to the study of such stationary measures on the circle and in analogous contexts.

Thursday, 08 August 2024

Let Sg be a closed surface of genus g ≥ 2. The curve graph corresponding to Sg, denoted by C(Sg), is a 1-dimensional simplicial complex whose vertices are isotopy classes of essential closed curves on Sg and two vertices share an edge if they represent mutually disjoint curves. Little is known about curves which are at a distance n ≥ 4 apart in C(Sg). This is primarily because the local infinitude of the vertices in C(Sg) hinders the calculation of distances in C(Sg).

In this talk, we will look at a family of pairs of curves on Sg which are at a distance 4 apart in C(Sg). These curves are created using Dehn twists. As an application, we will deduce an upper bound on the minimal intersection number of curves at a distance 4 apart in C(Sg). Finally, we will look at an example of a pair of curves on S2 which are at a distance 5 apart in C(S2).

In this talk, we will introduce the notion of an Anosov representation and explain how it can be used to give an equivalent criterion for proper affine actions.