** Large scale geometry of large groups **

**1. Motivation **

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 1Say 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.

**Examples**.

- 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 2Say 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 .

**Examples**.

- Compactly generated groups.
- groups.
- Infinite dimensional Banach spaces as additive groups.
- Rosendal: where is a regular tree with counable degree (vertex stabilizer is ).

**Non-example**.

- product of infinitely many copies of .

**3. Result **

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 .

**4. Proof **

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 .