Thomaes Haettel - Group actions on Helly graphs and injective metric spaces - I-III

Abstract

We will discuss Helly graphs and injective metric spaces: basic definitions, elementary properties, simple examples. We will focus on nonpositive curvature features, such as local properties, bicombings and classification of isometries. We will present numerous groups acting geometrically by isometries on such spaces, such as hyperbolic groups, braid groups, mapping class groups and lattices in Lie groups.

Talk

The notes to the talk can be found here.

Teaser:

• Are hyperbolic groups CAT(0)? Open!
• Are braid groups CAT(0)? Open!
• Are CAT(0) groups biautomatic? No!

Well we don’t know the answers to the first two but we know:

• Hyperbolic groups are Helly
• Braids are Helly
• Helly groups are biautomatic.

I Definition

Definition

$X$ is a Hyperconvex space if some complex things are fulfilled.

Remark: If $X$ is geodesic, then $X$ is hyperconvex if balls in $X$ satisfy the Helly property, i.e. pairwise intersecting balls globally intersect.

Definition

$X$ is an Injective metric space if for all $A\subseteq B$ matrix spaces, any 1-Lipschitz map $l:A\to X$ extends to a 1-Lipschitz map $\bar{l}:B\to X$.

Finally an explanation for injective spaces!

Definition

$X$ is an absolute 1-Lipschitz retract if for every isometric embedding $X\to Y$, there exists a 1-Lipshitz retraction $Y\to X$ (?)

Theorem

$X$ is hyperconvex iff. $X$ is injective iff. $X$ absolute $1$-Lipschitzretract.

Example

• $\mathbb{R}$ is hyperconvex, it is injective
• $({\mathbb{R}}^2,l^2)$ is not injective
• $({\mathbb{R}}^2,l^{\infty })$ is injective

Exercise:

• Disprove hyperconvexity in $({\mathbb{R}}^3,l^1)$.
• $l^{\infty }$ products of injective spaces are injective.
• Complete $\mathbb{R}$-trees are injective.
• Any finite-dimensional CAT(0) cube complex

Proposition

For any set $X$, $l^{\infty }(X,\mathbb{R})$ is injective.

Proposition

Any injective metric space is Contractible space and Geodesic space and Complete space

Example

Complete $\mathbb{R}$-trees are injective.

Definition

A connected graph is a Helly graph if pairwise intersecting balls globally intersect.

Example

• A line is Helly
• Trees are Helly
• Complete graphs are Helly

3. Group actions

Definition

A group is called injective or Helly group if it acts properly and cocompactly by isometries on an Injective metric space or Helly graph.

We have Helly implies injective.

Example

• Free abelian groups are Helly (act on $\mathbb{R}$)
• Free groups are Helly (act on trees)
• Hyperbolic group are Helly (Act on hyperbolic spaces)
• Braid group (and FC type Artin group and Garside group) are Helly
• Cocompact lattices in $G{L}_n(\mathbb{Q})$ are Helly
• Cocompact lattices in $G{L}_n(R)$ are injective.
• Mapping class groups act properly and coboundedly on an injective space. This makes them injective.
• $Out(F_n)$ for $n\ge 3$ is not injective. (Outer automorphisms of free group)

Some properties

If $G$ is Helly

• $G$ has type $\underline{F}$
• $G$ is a Biautomatic group

II Hulls

1. Injective Hull

Why are there so many groups acting on injective metric spaces?

Theorem

Any metric space $X$ embeds isometrically in a unique minimal injective metric space, its Injective hull $E(X)$.

We describe the explicit model for the injective hull $E(X)$. Let $\Delta (X)$ be all function from $X$ to ${\mathbb{R}}^{+}$ where the sum of the images if bigger then the distance of the preimage. Now we construct $E(X)$ as a minimal version of $\Delta (X)$.

Example

Let $X$ be the space of two points $a,b$ with distance $1$. Then $E$ will become the segment between $(1,0)$ and $(0,1)$.

We draw how the injective hull look like for $3$ points or $4$ points or $5$ points.

We study ${\mathbb{R}}^2$ with the hexagonal norm (the norm, where the norm ball is a hexagon). This is isometric to $({\mathbb{R}}^3,l^{\infty })$. But what is the injective hull of the Hyperbolic plane.

2. Helly Hull

Theorem

Any connected graph $X$ embeds isometrically in a unique minimal Helly graph, the Helly graph $H(X)$.

Theorem

$X$ connected graph, then $H(X)=E(X)\cup {\mathbb{N}}^X$. Meaning, we throw away all non-integer points.

Example

A square is a simple graph that is not Helly. To make it Helly, we can add diagonal or else we would shorten the distance of diagonals. Instead we enter a central vertex and connect to the corners. Same for the pentagon.

3. Orthosimplices

Definition

The standard $k$-orthosimplex is the simplex in ${\mathbb{R}}^k$ with ordered vertices $v_0=(0,...,0),v_1=(1,0,...,0),v_2=(1,1,0,...,0),...,v_k=(1,...,1)$ The $l^{\infty }$ orthocomplex metric is the ordered $l^{\infty }$-matrix. I can’t read the rest :/

Definition

In a graph a … clique is a complexe subgraph which is an intersection of balls.

Theorem

$X$ a Helly graph, $P_X$ the poset of cliques. Then $E(X)$ is $2$-isometric to $(|P_x|,l^{\infty })$

This gives a recipe to describe the injective hull of a graph. Under some assumtion we can use $E(X)=E(H(X))$.

Theorem (Long)

Let $X$ be a hyperbolic locally finite graph. Then $H(X)$ is at distance $\le \delta$ from $X$ and $H(X)$ is locally finite.

As a corollary hyperbolic graphs are Helly.

3. Taming the injective hull

Theorem

Let $X$ be a hyperbolic graph. Then $H(X)$ is hyperbolic and the embedding of $X$ in $H(X)$ is dense.

Definition

A connected graph $X$ has stable intervals if all geodesic intervals from a vertex $x$ fellow travel to any two adjacent vertices $y,z$. (Meaning the geodesic paths are contained in a $\delta$-neighbourhood for a global $\delta$)

Example

• Helly graphs
• Median graphs
• Hyperbolic graphs

Theorem

$X$ be a locally finite connected graph with stable intervals. Then $H(X)$ is locally finite.

Corollary

Let $G$ be a Hyperbolic group. $X$ a Cayley-Graph. Then the action of $G$ on $X$ is cocompact.

III NPC Properties

1. Bicombing

Definition

A (conical, geodesic) Bicombing on a metric space is a map $\sigma :X\times X\times [0,1]\to X$such that.

1. Between any two points there exists a constant speed geodesic parametrised by $\sigma (x,y,t)$
2. Any pair of geodesics between four points metrically curve towards another, not away from another.

Theorem (Long)

Any injective metric space $X$ admits an $Isom(X)$-inv. equivariant bicombing.

Corollary

Injective group have quadratic Dehn-Funktion.

This is not hyperbolic. But it is interesting

2. Fixed points

Theorem (Long)

$X$ injective space and $G$ acting on $X$ with bounded orbits. Then the fixed point set is non empty and the set of fixed points is injective.

3. Local to global

Definition

A connected graph $X$ is $1$-Helly if pairwise intersecting balls of radius 1 globally intersect.

Theorem

$X$ connected graph is Helly if

1. Some complex of $X$ is simply connected
2. $X$ is $1$-Helly

Theorem

A metric space is injective iff

1. $X$ is simply connected
2. $X$ is locally injective

IV Lattices