Discrete Isometry Group of Higher Rank Symmetric Spaces (Lecture - 01) by Misha Kapovich

Geometry, Groups and Dynamics (GGD) - 2017

DATE: 06 November 2017 to 24 November 2017

VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru

The program focuses on geometry, dynamical systems and group actions. Topics are chosen to cover the modern aspects of these areas in which research has been very active in recent times. The program will have two parts, a pre­school in the first week and an advanced school during the remaining two weeks. The pre­-school will be devoted to covering the necessary background material for the advanced school during the remaining two weeks. The courses for the first week of pre-school include:

Discrete subgroups of Lie groups, by Pralay Chatterjee (IMSc, Chennai )
Introduction to hyperbolic geometry, by Subhojoy Gupta (IISc, Bangalore)
Introduction to Geometric Group Theory, by Pranab Sardar (IISER, Mohali)
Crash course on Riemannian geometry, by Harish Seshadri (IISc, Bangalore)
Teichmüller Theory, old and new, by Athanase Papadopoulos (Univ. Strasbourg, France)
More details of the courses will be given soon on this webpage.

The advanced school will consist of courses aimed at graduate students and young researchers who are either working or want to work in the broad theme of the program. The courses will be supplemented by a few research level surveys and lectures by some eminent experts in the field. These supplementary lectures are meant to connect the topics of the advanced school to cutting edge research to open new horizons of research for the participants of the school.

The following courses and mini courses for the two-week advanced school have been planned.

Riemann Surfaces and the Absolute Galois Group, by Norbert A'Campo (Univ. Basel, Switzerland)
Hierarchically hyperbolic space, by Jason Behrstock (CUNY, USA)
Projection Complexes, by Mladen Bestvina (Univ. Utah, USA)
Geodesic flow of the Weil­Petersson metric, by Keith Burns (Northwestern Univ., USA)
Growth in groups, by Rostislav Grigorchuk (TAMU, USA)
Higher Teichmüller Theory, by Francois Labourie (Univ. Nice, France) and Misha Kapovich (Univ. Davis, USA)
Boundary theory of hyperbolic groups, by Mahan Mj. (TIFR, India)
Geometrically infinite Kleinian groups, by Ken'ichi Ohshika (Osaka Univ., Japan)
Teichmüller Theory, old and new, by Athanase Papadopoulos (Univ. Strasbourg, France)
Deformation of complex hyperbolic lattices, by Pierre Will (Univ. Grenoble, France)
There will be some expository lectures as well. The program is followed by ICTS discussion meeting : Surface Group Representations and Geometric Structures.

0:00:00 Geometry, Groups and Dynamics (GGD) - 2017
0:00:08 Discrete Isometry Group of Higher Rank Symmetric Spaces (Lecture - 01)
0:00:55 Goal
0:04:31 Historically, the theory of discrete subgroups
0:13:35 (1) High homogeneity
0:16:18 (2) Cartan decomposition
0:18:25 (3) Convergence property
0:21:51 Convergence is uniform on compacts
0:28:38 Exercise
0:29:37 Definition (Convergence action)
0:33:03 (4) H In is hyperbolic
0:35:27 (5) Ahandance of convex subsets
0:42:19 Now to higher rank
0:48:12 Congruence invariants of geodesic segments
0:55:53 Definition
0:58:02 Exercise - Relate d delta (x, y) to d delta (y, x)
1:16:36 Problem
1:18:21 Theorem (Aun Parrean)