Geometric group theory

A covering graph is a graph map that induces a bijection on the incident oriented edges at every vertex. Connected based covers of a graph correspond to subgroups of its fundamental group.

Schreier coset graph

For

H \leq G

and a generating set

S

, the Schreier coset graph has vertices

H g

and a directed

s

-edge from

H g

H g s

for every

s \in S

. For a free group, this is the covering graph associated with

H

Core graph of a subgroup

The core of a based covering graph is the union of all reduced loops through its base vertex. A subgroup of a finitely generated free group is finitely generated exactly when its core graph is finite.

Bass-Serre theory

Bass-Serre theory describes group splittings as actions on trees and reconstructs a graph of groups from the stabilizers of such an action.

Bass-Serre tree ( $T$ )

For

G = A *_{C} B

, the Bass-Serre tree has vertices

G / A ⊔ G / B

and edges

G / C

, with

g C

joining

g A

g B

. Vertex degrees are the indices of the corresponding images of

C

HNN extension

Given isomorphic subgroups

C, D \leq A

and an isomorphism

ϕ : C \to D

, the HNN extension is

⟨ A, t ∣ t c t^{- 1} = ϕ (c) for every c \in C ⟩ .

Its stable letter

t

records the loop edge in the associated graph of groups.

Baumslag-Solitar group ( $BS (m, n)$ )

The Baumslag-Solitar group is

BS (m, n) = ⟨ a, t ∣ t a^{m} t^{- 1} = a^{n} ⟩ .

Normal form theorem for an amalgamated free product

A *_{C} B

, an alternating product of elements of

A ∖ C

and

B ∖ C

is nonidentity. More generally, choosing coset representatives gives a unique reduced normal form.

Klein bottle group

The fundamental group of the Klein bottle has presentation

⟨ a, b ∣ ab a^{- 1} = b^{- 1} ⟩ .

Its subgroup

⟨ a^{2}, b ⟩ ≅ Z^{2}

has index two.

Group action on a tree

A group action on a tree is a homomorphism to the tree's combinatorial isometry group. Barycentric subdivision removes edge inversions without changing the essential action.

Isometry of a tree

Every inversion-free tree isometry is either elliptic, fixing a vertex, or hyperbolic, translating along a unique bi-infinite geodesic.

Elliptic isometry of a tree

An elliptic tree isometry fixes a vertex. Its fixed-point set is a nonempty convex subtree.

Hyperbolic isometry of a tree

A hyperbolic tree isometry has positive translation length and preserves a unique bi-infinite geodesic.

Axis of a tree isometry ( $A_{g}$ )

The axis of a hyperbolic tree isometry

g

is its unique invariant line. The isometry acts on this line by translation through its translation length.

Translation length ( $ℓ (g)$ )

The translation length of an isometry

g

of a metric space is

ℓ (g) = in f_{x} d (x, gx)

. For a hyperbolic tree isometry, it is attained exactly on the axis of a tree isometry.

Serre lemma for tree actions

If two elliptic isometries

g, h

of a tree have elliptic product

g h

, then their fixed subtrees intersect.

Helly property

Finite families of convex subtrees of a tree have the Helly property: if every pair intersects, then their total intersection is nonempty.

Fuchsian group

A Fuchsian group is a discrete subgroup of

PSL_{2} (R)

, acting by orientation-preserving isometries of the hyperbolic plane. It is non-elementary when it has no finite orbit in the hyperbolic plane or its ideal boundary.

Parabolic isometry of the hyperbolic plane

A nonidentity orientation-preserving isometry of the hyperbolic plane is parabolic when it has one fixed point on the ideal boundary and none in the plane. In the upper half-plane model, every parabolic isometry is conjugate to a nonzero horizontal translation.

Hyperbolic triangle group ( $Δ (p, q, r)$ )

When

1/ p + 1/ q + 1/ r < 1

, the orientation-preserving hyperbolic triangle group has presentation

Δ (p, q, r) = ⟨ a, b, c ∣ a^{p} = b^{q} = c^{r} = ab c = 1 ⟩

and is a non-elementary Fuchsian group.

Quasi-isometry

A map

f : X \to Y

is a

(λ, ε)

-quasi-isometric embedding when

λ^{- 1} d_{X} (x, x^{'}) - ε \leq d_{Y} (f (x), f (x^{'})) \leq λ d_{X} (x, x^{'}) + ε .

It is a quasi-isometry when every point of

Y

lies a uniformly bounded distance from its image.

Quasi-isometric embedding

A quasi-isometric embedding satisfies the two-sided coarse distance inequality in the definition of quasi-isometry, without requiring its image to be coarsely dense.

Quasi-isometrically embedded subgroup

A finitely generated subgroup

H \leq G

is quasi-isometrically embedded when its inclusion, equipped with word metrics from finite generating sets, is a quasi-isometric embedding. This property is independent of those generating sets.

Bilipschitz equivalence

A bilipschitz equivalence distorts all distances by multiplicative constants bounded away from zero and infinity. Every bilipschitz equivalence is a quasi-isometry.

Quasigeodesic

A quasigeodesic is a quasi-isometric embedding of an interval into a metric space.

Morse lemma for quasi-geodesics

In a Gromov-hyperbolic geodesic metric space, every quasigeodesic segment stays within a uniformly bounded Hausdorff distance of a geodesic segment with the same endpoints. The bound depends only on the hyperbolicity and quasigeodesic constants.

Gromov-hyperbolic metric space ( $δ$ )

A geodesic metric space is

δ

-hyperbolic when each side of every geodesic triangle lies in the closed

δ

-neighbourhood of the other two sides.

Hyperbolic group

A finitely generated group is hyperbolic when one, equivalently every, Cayley graph for a finite generating set is a Gromov-hyperbolic metric space.

Quasi-tree

A quasi-tree is a geodesic metric space quasi-isometric to a tree.

Bottleneck property

A geodesic metric space has the bottleneck property if every path joining the endpoints of a geodesic passes within one uniform distance of that geodesic's midpoint. A geodesic metric space is a quasi-tree exactly when it has this property.

Quasiconvex subset

A subset

Y

of a geodesic metric space is

K

-quasiconvex when every geodesic between points of

Y

lies in the closed

K

-neighborhood of

Y

Quasiconvex subgroup

A subgroup is quasiconvex when it is a quasiconvex subset of a Cayley graph. In a hyperbolic group, a finitely generated subgroup is quasiconvex exactly when it is quasi-isometrically embedded.

Displacement function ( $d_{g} (x)$ )

For an isometry

g

of a metric space, its displacement function is

d_{g} (x) = d (x, gx)

. It is

2

-Lipschitz.

Proper metric space

A metric space is proper when every closed bounded subset is compact.

Cocompact group action

A group action on a space is cocompact when its quotient is compact, equivalently when some compact set has translates covering the space.

Properly discontinuous group action