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The Svarc-Milnor lemma

Statement

Let $G$ act geometrically on a geodesic metric space $X$ , fix $x_{0} \in X$ , and choose $D \geq 0$ such that every point of $X$ lies within $D$ of $G \cdot x_{0}$ . Then $G$ is finitely generated. More precisely, if $S : = {g \in G : d_{X} (x_{0}, g \cdot x_{0}) \leq 2 D + 1}$ is the finite generating set obtained from Cobounded proper geodesic actions produce finite generating sets, then the orbit map $ϕ_{x_{0}} : (G, d_{S}) ⟶ X, ϕ_{x_{0}} (g) : = g \cdot x_{0},$ is a quasi-isometry.

Facts & Assumptions

Given: A geometric action of $G$ on a geodesic metric space $X$ , a point $x_{0} \in X$ , and a real $D \geq 0$ such that every point of $X$ lies within distance at most $D$ of the orbit $G \cdot x_{0}$ .

[L1]

The set $S : = {g \in G : d_{X} (x_{0}, g \cdot x_{0}) \leq 2 D + 1}$ is a finite generating set of $G$ (Cobounded proper geodesic actions produce finite generating sets).

[L2]

For this generating set, the orbit map satisfies $d_{X} (g \cdot x_{0}, h \cdot x_{0}) \leq M d_{S} (g, h)$ for some constant $M \geq 0$ (Orbit maps of isometric actions are coarse Lipschitz).

[L3]

A subset is coarsely dense when every point of the space lies within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

[L4]

The word metric is $d_{S} (g, h) = ∣ g^{- 1} h ∣_{S}$ (The word metric of a group with respect to a generating set).

Proof

technique · direct

1.1L1L2L4

By [L1], the set $S$ is finite and generates $G$ , so $d_{S}$ is a word metric on $G$ . Step [L2] gives the coarse-Lipschitz upper bound for $ϕ_{x_{0}}$ .

1.2L3given

The orbit $ϕ_{x_{0}} (G) = G \cdot x_{0}$ is $D$ -dense in $X$ by the choice of $D$ , so it is coarsely dense in the sense of [L3].

1.3L1L4algebra

For $g \in G$ , the proof of [L1] writes $g$ as a product of at most $m$ elements of $S$ , where $m$ is the least positive integer with $d_{X} (x_{0}, g \cdot x_{0}) \leq m$ . Therefore $∣ g ∣_{S} \leq d_{X} (x_{0}, g \cdot x_{0}) + 1$ . Applying this to $g^{- 1} h$ and using isometricity gives $d_{S} (g, h) = ∣ g^{- 1} h ∣_{S} \leq d_{X} (g \cdot x_{0}, h \cdot x_{0}) + 1$ .

2.1L1step 1.2construct

The finite generating set $S$ gives a fixed enumeration of $G$ : order words in the finite alphabet $S$ first by length and then lexicographically, and retain the first word representing each group element. The empty word represents the identity. For each $x \in X$ , step 1.2 makes the set of words whose represented element $g$ satisfies $d_{X} (x, g \cdot x_{0}) \leq D$ nonempty. Let $r (x)$ be the element represented by its first word. This defines one function $r : X \to G$ without choosing independently over $X$ , and $d_{X} (x, r (x) \cdot x_{0}) \leq D$ for every $x$ .

3.1step 1.3step 2.1algebra

For $x, y \in X$ , step 1.3 with $g = r (x)$ and $h = r (y)$ gives $d_{S} (r (x), r (y)) \leq d_{X} (r (x) \cdot x_{0}, r (y) \cdot x_{0}) + 1 \leq d_{X} (x, y) + 2 D + 1.$ So $r : X \to G$ is coarse Lipschitz.

3.2step 1.3step 2.1algebra

For every $x \in X$ , step 2.1 gives $d_{X} (ϕ_{x_{0}} (r (x)), x) \leq D$ . For every $g \in G$ , step 1.3 and step 2.1 with $x = g \cdot x_{0}$ give $d_{S} (r (g \cdot x_{0}), g) \leq d_{X} (r (g \cdot x_{0}) \cdot x_{0}, g \cdot x_{0}) + 1 \leq D + 1.$ Thus $r \circ ϕ_{x_{0}}$ and ${id}_{G}$ , and also $ϕ_{x_{0}} \circ r$ and ${id}_{X}$ , are at bounded distance.

4.1L3step 1.1step 3.1step 3.2∎

Step 1.1 shows that $ϕ_{x_{0}}$ is coarse Lipschitz, and step 3.2 gives a coarse Lipschitz quasi-inverse. Hence the orbit map is a quasi-isometry by [L3].

Depends on

Used by

Groups acting geometrically on the same space are quasi-isometric Corollary
Closed surface groups are hyperbolic Example
Free groups act geometrically on regular trees Example
The integers act geometrically on the real line Example
Zⁿ acts geometrically on Euclidean n-space Example

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. Löh, Geometric Group Theory, Sections 4.4 and 5.1 (standard reference, not scraped)
C. Drutu and M. Kapovich, Lectures on Geometric Group Theory, Chapter 5 (standard reference, not scraped)