## Notes of Kathryn Mann’s Cambridge lecture 11-05-2017

Large scale geometry of large groups

1. Motivation

Groups acting on manifolds. ${Hom(G,Homeo(M))}$ or ${Hom(G,Diff(M))}$ seem to be inaccessible, except if ${M=S^1}$. I make an attempt to quasify ${Homeo_+(S^1)}$.

Poincare: the rotation number ${rot:Homeo_+(S^1)\rightarrow {\mathbb R}/{\mathbb Z}}$ descends from ${\widetilde{rot}:Homeo_{\mathbb Z}({\mathbb R})\rightarrow {\mathbb R}}$ defined by

$\displaystyle \begin{array}{rcl} \widetilde{rot}(\tilde f)=\lim_{n\rightarrow\infty}\frac{\tilde f^n(x)-x}{n}. \end{array}$

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 ${Homeo_+(S^1)}$ is determined up to semi-conjugacy by the rotation numbers of a set of generators for the group and ${\widetilde{rot}(\tilde g_1)+\widetilde{rot}(\tilde g_2)-\widetilde{rot}(\tilde g_1\tilde g_2)}$.

Rotation numbers can be viewed as coordinates on. They also detect connected components of ${Hom(\Gamma,Homeo_+(S^1)}$

1.2. It sees the geometry of subgroups

It is easy to see that ${\widetilde{rot}(\tilde f)=0}$ iff ${\tilde f}$ is distorted in some finitely generated subgroup of ${Homeo_{\mathbb Z}({\mathbb R})}$.

2. Large scale geometry of non locally compact groups

Our dream is to quasify ${Homeo_0(M)}$ for other manifolds ${M}$.

2.1. Early hints: boundedness

Definition 1 Say topological group ${G}$ has property ${(OB)}$ 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: ${\mathfrak{S}_{\mathbb N}}$, even with discrete topology.
• Calegari-Freedman-Cornulier: ${Homeo_0(S^n)}$ with discrete topology.
• Le Roux-Mann : ${Diff({\mathbb R}^n)}$ with discrete topology.

Non-examples. ${Diff(M)}$, ${M}$ compact, admits ${C^0}$ metric.

Definition 2 Say a subset ${A\subset G}$ is ${(OB)}$ if bounded i every compatible left-invariant metric. Say ${G}$ is locally ${(OB)}$ if there is an ${(OB)}$ neighborhood of 1.

Theorem 3 (Rosendal) Let ${G}$ be a separable metrizable group. If ${G}$ is locally ${(OB)}$ and ${(OB)}$-generated, then

1. All ${(OB)}$ generating sets define quasi-isometric word metrics.
2. Every ${d'}$ a compatible left-invariant metric on ${G}$ is bounded above by such a word metric.
3. There exists a compatible metric which is quasi-isometric to word metrics.

Here is a useful characterization. ${A\subset G}$ has ${(OB)}$ iff there exists a neighborhood ${U}$ of 1, an integer ${n}$ ad finite set ${F\subset G}$ such that ${A\subset (FU)^n}$.

Examples.

• Compactly generated groups.
• ${(OB)}$ groups.
• Infinite dimensional Banach spaces as additive groups.
• Rosendal: ${Aut(T)}$ where ${T}$ is a regular tree with counable degree (vertex stabilizer is ${(OB)}$).

Non-example.

• ${{\mathbb R}^\infty=}$ product of infinitely many copies of ${{\mathbb R}}$.

3. Result

Theorem 4 (Mann-Rosendal) ${M}$ be a compact topological manifold. Then ${Homeo_0(M)}$ is locally ${(OB)}$ and ${(OB)}$-generated.

If furthermore ${\pi_1(M)}$ is infinite, ${Homeo_0(M)}$ is unbounded.

In fact, it contains a coarsely embedded copy of ${C^0[0,1]}$.

Theorem 5 (Cohen) ${Diff^r([0,1])}$, ${1\leq r<\infty}$, is locally ${(OB)}$.

${Diff^1([0,1])}$ is quasi-isometric to ${C^0([0,1])}$.

4. Proof

Fragmentation. Any homeo close enough to identity is a product of homeos with small supports.

This implies local ${(OB)}$. 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 ${Homeo_+(S^1)}$

${Homeo_{\mathbb Z}({\mathbb R})}$ is quasi-isometric to ${{\mathbb R}}$. More generally, there is an exact sequence ${1\rightarrow\pi_1(M)\rightarrow G\rightarrow Homeo_0(M)\rightarrow 1}$ where ${G}$ are lifts to ${\tilde M}$. This is not quasi-isometric to a product, in general. There is a central subgroup ${A<\pi_1(M)}$

Theorem 6 (Mann-Rosendal) If there is a section ${\pi_1(M)/A\rightarrow \pi_1(M)}$ which is a quasi-isometric embedding, then this gives a quasi-isometry ${G\sim Homeo_0(M)\times\pi_1(M)}$.