Large scale geometry of large groups
Groups acting on manifolds. or seem to be inaccessible, except if . I make an attempt to quasify .
Poincare: the rotation number descends from defined by
This is a quasimorphism.
This is a good quasification in the following two senses.
1.1. No loss of information
Ghys-Matsumoto show that a subgroup of is determined up to semi-conjugacy by the rotation numbers of a set of generators for the group and .
Rotation numbers can be viewed as coordinates on. They also detect connected components of
1.2. It sees the geometry of subgroups
It is easy to see that iff is distorted in some finitely generated subgroup of .
2. Large scale geometry of non locally compact groups
Our dream is to quasify for other manifolds .
2.1. Early hints: boundedness
Definition 1 Say topological group has property is every compatible left-invariant metric is bounded.
OB comes from the equivalent definition that every continuous isometric action on a metric space has bounded orbits.
- Compact groups.
- Galvin: , even with discrete topology.
- Calegari-Freedman-Cornulier: with discrete topology.
- Le Roux-Mann : with discrete topology.
Non-examples. , compact, admits metric.
Definition 2 Say a subset is if bounded i every compatible left-invariant metric. Say is locally if there is an neighborhood of 1.
Theorem 3 (Rosendal) Let be a separable metrizable group. If is locally and -generated, then
- All generating sets define quasi-isometric word metrics.
- Every a compatible left-invariant metric on is bounded above by such a word metric.
- There exists a compatible metric which is quasi-isometric to word metrics.
Here is a useful characterization. has iff there exists a neighborhood of 1, an integer ad finite set such that .
- Compactly generated groups.
- Infinite dimensional Banach spaces as additive groups.
- Rosendal: where is a regular tree with counable degree (vertex stabilizer is ).
- product of infinitely many copies of .
Theorem 4 (Mann-Rosendal) be a compact topological manifold. Then is locally and -generated.
If furthermore is infinite, is unbounded.
In fact, it contains a coarsely embedded copy of .
Theorem 5 (Cohen) , , is locally .
is quasi-isometric to .
Fragmentation. Any homeo close enough to identity is a product of homeos with small supports.
This implies local . Indeed, every homeo close to identity is then a product of conjugates of homeos with support in the same small open set.
5. Back to
is quasi-isometric to . More generally, there is an exact sequence where are lifts to . This is not quasi-isometric to a product, in general. There is a central subgroup
Theorem 6 (Mann-Rosendal) If there is a section which is a quasi-isometric embedding, then this gives a quasi-isometry .