PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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A subgroup of finite index in a finitely generated group is finitely generated, and its inclusion is a quasi-isometry

Statement

A subgroup of finite index in a finitely generated group is finitely generated, and its inclusion is a quasi-isometry.

Facts & Assumptions

Given: The hypotheses of the Statement.

[F1]

A group is finitely generated when some finite subset generates it (Finitely generated groups).

[L1]

The word length $∣ g ∣_{S}$ is the least $n$ such that $g$ is a product of $n$ elements of $S \cup S^{- 1}$ (Word length of a group element with respect to a generating set).

[L2]

Word length is defined on every element and satisfies $∣ g h ∣_{S} \leq ∣ g ∣_{S} + ∣ h ∣_{S}$ , $∣ g^{- 1} ∣_{S} = ∣ g ∣_{S}$ , and $∣ g ∣_{S} = 0$ exactly when $g$ is the identity (Word length is defined on every element and satisfies the subadditivity, inversion and vanishing laws).

[L3]

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

[L4]

A map is $(L, C)$ -coarse Lipschitz when $d (f (x), f (x^{'})) \leq L d (x, x^{'}) + C$ , and an $(L, C)$ -quasi-isometric embedding when in addition $L^{- 1} d (x, x^{'}) - C \leq d (f (x), f (x^{'}))$ (Coarse Lipschitz maps and quasi-isometric embeddings).

[L5]

A subset is coarsely dense when every point of the space is 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).

[L6]

$[G : H] : = ∣ G / H ∣$ (The coset set $G / H$ and the index $[G : H]$ of a subgroup).

[L7]

$g H : = {g h : h \in H}, H g : = {h g : h \in H} .$ (Left and right cosets $g H$ and $H g$ of a subgroup).

[L8]

For a subgroup $H \leq G$ , the set of distinct left cosets ${g H : g \in G}$ is a partition of $G$ : every element belongs to a left coset, every coset is nonempty, and two left cosets are either equal or disjoint. (The left cosets of a subgroup partition the group).

[L9]

A set $A$ is finite when $A \approx n$ for some $n \in N$ . (The cardinality $∣ A ∣$ of a finite set).

Proof

technique · direct

1.1

Fix a finite left transversal $T$ for $H$ in $G$ that contains the identity; the cosets partition the group and there are finitely many of them.

F1L6L7L8L9choose

2.1

Rewriting a product of ambient generators by pushing transversal representatives to the right expresses each subgroup element in the finite set of products that fall back into the subgroup, so that set generates it.

F1L1L2step 1.1

3.1

Write each $g \in G$ uniquely as $g = r (g) t (g)$ with $r (g) \in H$ and $t (g) \in T$ . If $h = g s$ with $s \in S \cup S^{- 1}$ and $g = r (g) t (g)$ , $h = r (h) t (h)$ , then $r (g)^{- 1} r (h) = t (g) s t (h)^{- 1} \in H$ is one of the finitely many subgroup elements used in step 2.1; so $d_{H} (r (g), r (h)) \leq 1$ in the word metric of $H$ for the generating set built there. Chaining along a shortest $S$ -expression gives $d_{H} (r (g), r (h)) \leq d_{G} (g, h)$ , so $r : G \to H$ is Lipschitz. Since $r$ is the identity on $H$ and every $g = r (g) t (g)$ lies within $\max_{t \in T} ∣ t ∣_{S}$ of the inclusion of $r (g)$ , the composites $r \circ i$ and $i \circ r$ are the identity on $H$ and at bounded distance from the identity on $G$ respectively. Therefore the inclusion $i : H ↪ G$ is a quasi-isometry with quasi-inverse $r$ .

L1L3L4L5step 1.1step 2.1∎

Depends on

Used by

The subgroup 2ℤ×ℤ has index two in ℤ² and its inclusion is a quasi-isometry Example

Dependency tree · two levels

32 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. Loh, Geometric Group Theory: An Introduction (2015 course version), 264 pp. (standard reference, not scraped)
C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)