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)}.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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