## Young Geometric Group Theory Meeting

### Young Geometric Group Theory Meeting

#### 9 - 13 January, 2012, Będlewo

The aim of the meeting was to bring together young researchers in geometric group theory, graduate students and post-docs, to allow them to learn from one another and from senior mathematicians invited to give tutorial courses in different branches of geometric group theory.

The conference was organized by Polish Academy of Sciences, University of Warsaw, University of Wroclaw, Adam Mickiewicz University and National Science Foundation.

During the conference, the following 4-hour topic courses were given:
Yves Benoist, Discrete subgroups of Lie groups,
Ruth Charney, Artin groups and their automorphisms,
Cornelia Drutu, Embeddings of groups,
Mark Feighn, Limit groups,

### Organizing Committee

• Kamil Duszenko (University of Wrocław)
• Piotr Przytycki (Polish Academy of Sciences)
• Paweł Zawiślak (University of Warsaw)
• Moon Duchin (Tufts University) – responsible for the coordination of the NSF funding.

### Scientific Committee

• Pierre-Emmanuel Caprace (Université catholique de Louvain)
• Rémi Coulon (Max-Planck-Institut, Bonn)
• Moon Duchin (Tufts University) - chair
• Tobias Hartnick (Université de Genève)
• Fanny Kassel (CNRS and Université Lille 1)
• John Mackay (University of Oxford)<
• Kevin Wortman (University of Utah)

### Programme

TUTORIAL SPEAKERS

Yves Benoist, Discrete subgroups of Lie groups Lecture 1 Lecture 2 Lecture 3 Lecture 4

This course is an introduction to Zariski dense subgroups of semisimple Lie groups. I will explain on a family of examples how very different approaches can be used to study these subgroups. These examples are the groups of projective transformations of the real projective space which preserve a properly convex set. The topics of the four talks will be:

1. Word hyperbolicity.
2. Zariski topology.
3. Moduli spaces.
4. Construction of examples.

Ruth Charney, Artin groups and their automorphisms Part 1 Part 2

1. An introduction to Artin groups.
2. Right-angled Artin groups.
3. Automorphisms of right-angled Artin groups.

Cornelia Druţu, Embeddings of groups Part 1 Part 2

The main theme of this tutorial course is that of embeddings of infinite groups, uniform, quasi-isometric, equivariant and non-equivariant, and the structures that can be induced on groups via such embeddings. It will cover the following topics:

1. relatively hyperbolic and CAT(0) groups,
2. products of (quasi-)trees, median spaces, L^p spaces,
3. property (T) and a-T-menability, Hilbert and Banach compression - families of graphs and embeddings.

Mark Feighn, Limit groups

This course will cover some results about limit groups, a class of groups that arises in problems at the intersection of geometric group theory and logic. Here is a concrete problem (solved by Kharlampovich-Myasnikov and Sela) that will be a focus of the course:

For a finitely generated group G, describe the set of homomorphisms from G to a free group.

It is perhaps surprising that this set has any structure at all. The main techniques will be JSJ-decompositions and groups acting on real trees. The course will also include a tutorial on these topics and is based on the joint paper with Mladen Bestvina Notes on Sela's work: Limit groups and Makanin-Razborov diagrams.

JUNIOR SPEAKERS

Michael Björklund, Limit theorems for random walks on groups

I intend to cover a few (classical and modern) approaches to quantitative limit theorems for the drift of random walks on metric groups; in particular limit theorems of central limit type. I will mostly focus on hyperbolic groups, but the methods are quite flexible and allow to give new proofs of a series of classical results.

Jing Tao, Diameter of the thick part of moduli space Page 1 Pages 2-7 Slides

Let S be a surface of genus g with p punctures with negative Euler characteristic. We study the diameter of the ε-thick part of moduli space of S equipped with the Teichmüller or Thurston's Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order log((g+p)/ε). The same result also holds for the ε-thick part of the moduli space of metric graphs of rank n equipped with the Lipschitz metric. The proof involves a sorting algorithm that sorts an arbitrary labeled tree with n labels with simultaneous Whitehead moves, where the number of steps is of order log(n). This is joint with Kasra Rafi.

Mathieu Carette, Spectral rigidity of automorphic orbits in the free group

Let F_n be the free group of rank n. Each point in Culler-Vogtmann's (unprojectivized) Outer Space cv_n is uniquely determined by its translation length function which assigns a positive real to each element of F_n. A subset X of F_n is spectrally rigid if each point in Outer Space is uniquely determined by the translation lengths of elements of X. I will discuss how Lipschitz distortion on Outer Space and currents on free groups are useful in finding sparse subsets of F_n which are spectrally rigid. This is joint work with Stefano Francaviglia, Ilya Kapovich, and Armando Martino.

Thomas Koberda, Right-angled Artin subgroups of right-angled Artin groups

I will present a systematic way of classifying all right-angled Artin subgroups of a given right-angled Artin group. The methods used have a number of corollaries: for instance, it can be shown that there is an embedding between a right-angled Artin group on a cycle of length m to one on a cycle of length n if and only if m=n+k(n-4) for some nonnegative integer k. I will also give some rigidity results. This is joint work with Sang-hyun Kim.

Johanna Mangahas, The geometry of right-angled Artin subgroups of mapping class groups

I'll describe joint work with Matt Clay and Chris Leininger on conditions for a finite set of mapping classes to generate a right-angled Artin group quasi-isometrically embedded in the mapping class group.

A locally compact group G has Kazhdan's property (T) if every action of G by affine isometries on a Hilbert space has a fixed point. In this talk we will be interested in strengthening this property by replacing the Hilbert space with other Banach spaces. In particular, we will show how to generalize the geometric/spectral method for proving property (T) to the setting of reflexive Banach spaces. We will also discuss examples of groups which satisfy our conditions for some L_p-spaces. These examples include random hyperbolic groups. We apply our results to give lower bounds on the conformal dimension of the boundary of such hyperbolic groups and to give quantitative estimates on vanishing of cohomology with coefficients in uniformly bounded representations. Other applications will also be discussed.

Patrick Reynolds, "Curve complex" for the free group

The introduction of the Culler-Vogtmann Outer space has ushered an adaptation to Out(F) of many useful aspects of the surface theory. Recently, this effort has witnessed substantial progress in adapting to Out(F) the Masur-Minsky hierarchy machine, which exploits the fact that the large-scale geometry of the Teichmüller space is coded in the geometry of the curve complex.

The aim of this talk is to highlight some difficulties that arise in adapting surface theory to the free group. Often complications arise from the fact that many structures associated to the free group and its automorphisms are morally "non-split," whereas structures associated to surfaces and their automorphisms are always morally "split." This phenomenon is well-illustrated by the fact that there are two curve complex analogues for Out(F).

### Short presentations by junior researchers

• Michael Bjorklund, Limit theorems for random walks on groups,
• Jing Tao, Diameter of the thick part of moduli space,
• Mathieu Carette, Spectral rigidity of automorphic orbits in the free group,
• Thomas Koberda, Right-angled Artin subgroups of right-angled Artin groups,
• Johanna Mangahas, The geometry of right-angled Artin subgroups of mapping class groups,
• Piotr Nowak, Poincare inequalities, rigid groups and applications,
• Patrick Reynolds, "Curve complex" for the free group

### Participants

1. Margarita Amchislavska (Cornell University)
2. Sylvain Arnt (Université d'Orléans)
3. Benjamin Beeker (Université Paris-Sud 11)
4. Antoine Beljean (Universität Münster)
5. Yves Benoist (Université Paris-Sud 11)
6. Michael Björklund (Eidgenössische Technische Hochschule Zürich)
7. Henry Bradford (University of Oxford)
8. Ayala Byron (Hebrew University)
9. Caterina Campagnolo (Université de Genève)
10. Mathieu Carette (Université catholique de Louvain)
11. Christopher Cashen (Université de Caen Basse-Normandie)
12. Ruth Charney (Brandeis University)
13. Corina Ciobotaru (Université catholique de Louvain)
14. Paweł Ciosmak (University of Warsaw)
15. Jan Czajkowski (University of Wrocław)
16. Dieter Degrijse (Katholieke Universiteit Leuven Campus Kortrijk)
17. Jonas Deré (Katholieke Universiteit Leuven Campus Kortrijk)
18. Aleksander Doan (University of Warsaw)
19. Maciej Dołęga (University of Wrocław)
20. Spencer Dowdall (University of Illinois)
21. Dennis Dreesen (Université catholique de Louvain)
22. Cornelia Druţu (University of Oxford)
23. Moon Duchin (University of Michigan)
24. Matthew Durham (University of Illinois)
25. Kamil Duszenko (University of Wrocław)
26. Ahmed Elsawy (Universität Düsseldorf)
27. Mark Feighn (Rutgers University)
28. Gregory Fein (Rutgers University)
29. Elizabeth Fink (University of Oxford)
30. Martin Fluch (Universität Bielefeld)
31. Światosław Gal (Universität Wien / University of Wrocław)
32. Łukasz Garncarek (University of Wrocław)
33. Ilya Gekhtman (University of Chicago)
34. Dominik Gruber (Universität Wien)
35. Jiyoung Han (Seoul National University)
36. Tobias Hartnick (Université de Genève)
37. Sebastian Hensel (Max-Planck Institut, Bonn)
38. David Hume (University of Oxford)
39. Alaa Jamal Eddine (Université d'Orléans)
40. Katarzyna Jankiewicz (University of Warsaw)
42. Pierre-Nicolas Jolissaint (Université de Neuchâtel)
43. Paweł Józiak (University of Wrocław)
44. Marek Kaluba (Adam Mickiewicz University)
45. Dawid Kielak (University of Oxford)
46. Sang-hyun Kim (Korea Advanced Institute of Science and Technology)
47. Thomas Koberda (Harvard University)
48. Juhani Koivisto (University of Helsinki)
49. Benno Kuckuck (University of Oxford)
50. Sang-hoon Kwon (Seoul National University)
51. Robin Lassonde (University of Michigan)
52. Adrien Le Boudec (Université Paris-Sud 11)
53. Jean Lécureux (Technion, Haifa)
54. Paul-Henry Leemann (Université de Genève)
55. Arie Levit (Hebrew University)
56. Joel Louwsma (University of Oklahoma)
57. John Mackay (University of Oxford)
58. Johanna Mangahas (Brown University)
59. Michał Marcinkowski (University of Wrocław)
60. Timothée Marquis (Université catholique de Louvain)
61. Alexandre Martin (Université de Strasbourg)
62. Sebastian Meinert (Freie Universität Berlin)
64. Piotr Nowak (Texas A&M University)
65. Tomasz Odrzygóźdź (University of Warsaw)
66. Damian Osajda (Universität Wien)
67. Hester Pieters (Université de Genève)
68. Thibault Pillon (Université de Neuchâtel)
69. Wojciech Politarczyk (Adam Mickiewicz University)
70. Maria Beatrice Pozzetti (Eidgenössische Technische Hochschule Zürich)
71. Tomasz Prytuła (University of Warsaw)
72. Janusz Przewocki (Polish Academy of Sciences)
73. Piotr Przytycki (Polish Academy of Sciences)
74. Doron Puder (Hebrew University)
75. Colin Reid (Université catholique de Louvain)
76. Patrick Reynolds (University of Utah)
77. Luis Manuel Rivera-Martinez (Universität Wien)
78. Moritz Rodenhausen (Universität Bonn)
79. Pascal Rolli (Eidgenössische Technische Hochschule Zürich)
80. Jordan Sahattchieve (University of Michigan)
81. Andrew Sale (University of Oxford)
82. Marco Schwandt (Universität Bielefeld)
83. Alessandro Sisto (University of Oxford)
84. Rizos Sklinos (Hebrew University)
85. Amit Solomon (Hebrew University)
86. Piotr Sołtan (University of Warsaw)
87. Aneta Stal (University of Warsaw)
88. Emily Stark (Tufts University)
89. Markus Steenbock (Universität Wien)
90. Harold Sultan (Columbia University)
91. Michał Szostakiewicz (University of Warsaw)
92. Jacek Świątkowski (University of Wrocław)
93. Krzysztof Święcicki (University of Warsaw)
94. Łukasz Świstek (University of Warsaw)
95. Jing Tao (University of Utah)
96. Samuel Taylor (University of Texas at Austin)
97. Ewa Tyszkowska (University of Gdańsk)
98. Richard Wade (University of Oxford)
99. Pei Wang (Rutgers University)
100. Sarah Wauters (Katholieke Universiteit Leuven Campus Kortrijk)
101. Christian Jens Weigel (Universität Gießen)
102. Jacek Wieszaczewski (University of Wrocław)
103. Stefan Witzel (Universität Münster)
104. Wenyuan Yang (Université Paris-Sud 11)
105. Gašper Zadnik (University of Ljubljana)
106. Matthew Zaremsky (Universität Bielefeld)
107. Paweł Zawiślak (University of Warsaw)
108. Joanna Zubik (Uniwersytet Wrocławski)