Geodesic triangles in the hyperbolic plane have the property that every point on a given side is at most distance two from one of the other sides, regardless of how big the triangle is. A Gromov hyperbolic metric space is a metric space with this thin triangles property. Amazingly, this simple definition leads to deep mathematics, particularly in relation to group theory. Gromov hyperbolic groups have good algebraic properties (for example, solvable word problem) and are "generic": if you pick a group at random, it will be Gromov hyperbolic.
Hyperbolic space \(\mathbb{H}^{n+1}\) has a sphere at infinity \(S^{n}\) which has a conformal structure. This was used, for example, in the original proof of Mostow rigidity. Likewise, a Gromov hyperbolic group \(X\) has a boundary at infinity \(\partial_\infty X\) which carries a canonical topological (in fact, conformal) structure. However, unlike the usual sphere, this boundary will often have fractal like properties.
The aim of this course is to give an introduction to Gromov hyperbolic groups and to explore some of the connections between the geometry of the boundary at infinity and the algebraic properties of the group.
Here is an outline of topics that we intend to cover.
Rough lecture notes will appear here as the course progresses.
January 19th: 0. Outline, 1. Gromov hyperbolic spaces, 2. Gromov hyperbolic groups, 3. Boundaries at infinity
January 26th: 3. cont., 4. The group of isometries of \( \mathbb{H}^n \) is the group of Mobius transformations of \(S^{n-1}\)
February 2nd: 4. cont., 5. Quasi-conformal and quasi-Mobius homeomorphisms.
February 9th: 5. cont., 6. Boundaries of Gromov hyperbolic groups and the correspondence between quasi-isometries of the space and quasi-Mobius homeomorphisms of the boundary.
February 16th: 6. cont., 7. Mostow rigidity, Rademacher's theorem.
February 23rd: 7. cont., 8. Sullivan, Tukia theorems.
March 1st: 8. cont., 9. Cohomology and topological dimension.
March 8th: 9. cont., 10. Conformal dimension.
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