Special RTG Lectures in Topology - Spring 2013

Michah Sageev

Technion Institute of Technology

Tue Jan 15 - Thu Jan 17 - Wed Jan 23 - Thu Jan 24

3:30 to 4:30 PM - RLM 9.166

CAT(0) cube complexes are spaces which enjoy geometric properties stemming from non-positive curvature, as well as certain combinatorial properties that give them the look and feel of “higher  dimensional trees.”

They have connections to various other topics, such as subgroup separability and Kazhdan's Property (T), and they played a central role in the work of Agol and Wise, culminating in the solution of The Virtual Haken Conjecture.

This lecture series will provide a general introduction to CAT(0) cube complexes and group actions on them.

Lecture 1: Basic notions. Following Gromov, we will describe a combinatorial condition for recognizing non-positively curvature for cube complexes. We will then go on to give examples and discuss hyperplanes and half-spaces, which give cube complexes their tree-like behavior.

Lecture 2: Cubulations.    We will describe how to construct CAT(0) cube complexes from pocsets spaces with walls, as well as give applications to small cancellation theory and 3-manifold.

Lecture 3: Hyperbolic-like behavior. CAT(0) spaces differ from negatively curved spaces in that they can contain flats. Consequently isometries can exhibit behavior which is different from a hyperbolic element acting on hyperbolic space. Nonetheless, it turns out that isometries often do exhibit such behavior. We will describe this behavior and describe how it yields The Tits Alternative Theorem. (This is joint work with Pierre-Emmanuel Caprace.)

Lecture 4: Special cube complexes. We will give a brief account of a subclass of non-positively curved cube complexes known as special cube complexes introduced by Haglund and Wise, and their connection to subgroup separability and The Virtual Haken Conjecture.