Commensurability (group theory)

In

.

Commensurability in group theory

Two

For example:

• A group is finite if and only if it is commensurable with the trivial group.
• Any two finitely generated free groups on at least 2 generators are commensurable with each other.^[2] The group SL(2,Z) is also commensurable with these free groups.
• Any two
  surface groups of genus
  at least 2 are commensurable with each other.

A different but related notion is used for subgroups of a given group. Namely, two subgroups Γ₁ and Γ₂ of a group G are said to be commensurable if the intersection Γ₁ ∩ Γ₂ is of finite index in both Γ₁ and Γ₂. Clearly this implies that Γ₁ and Γ₂ are abstractly commensurable.

Example: for nonzero real numbers a and b, the subgroup of R generated by a is commensurable with the subgroup generated by b if and only if the real numbers a and b are commensurable, meaning that a/b belongs to the rational numbers Q.

In geometric group theory, a finitely generated group is viewed as a metric space using the word metric. If two groups are (abstractly) commensurable, then they are quasi-isometric.^[3] It has been fruitful to ask when the converse holds.

There is an analogous notion in linear algebra: two linear subspaces S and T of a vector space V are commensurable if the intersection S ∩ T has finite codimension in both S and T.

In topology

Two

, commensurable spaces have commensurable fundamental groups.

Example: the Gieseking manifold is commensurable with the complement of the figure-eight knot; these are both noncompact hyperbolic 3-manifolds of finite volume. On the other hand, there are infinitely many different commensurability classes of compact hyperbolic 3-manifolds, and also of noncompact hyperbolic 3-manifolds of finite volume.^[4]

The commensurator

The commensurator of a subgroup Γ of a group G, denoted Comm_G(Γ), is the set of elements g of G that such that the conjugate subgroup gΓg⁻¹ is commensurable with Γ.^[5] In other words,

${\displaystyle \operatorname {Comm} _{G}(\Gamma )=\{g\in G:g\Gamma g^{-1}\cap \Gamma {\text{ has finite index in both }}\Gamma {\text{ and }}g\Gamma g^{-1}\}.}$

This is a subgroup of G that contains the

N_G(Γ) (and hence contains Γ).

For example, the commensurator of the

The abstract commensurator

The abstract commensurator of a group ${\displaystyle G}$, denoted Comm${\displaystyle (G)}$, is the group of equivalence classes of isomorphisms ${\displaystyle \phi :H\to K}$, where ${\displaystyle H}$ and ${\displaystyle K}$ are finite index subgroups of ${\displaystyle G}$, under composition.[7] Elements of ${\displaystyle {\text{Comm}}(G)}$ are called commensurators of ${\displaystyle G}$.

If ${\displaystyle G}$ is a connected

not isomorphic to ${\displaystyle {\text{PSL}}_{2}(\mathbb {R} )}$, with trivial center and no compact factors, then by the Mostow rigidity theorem, the abstract commensurator of any irreducible lattice ${\displaystyle \Gamma \leq G}$ is linear. Moreover, if ${\displaystyle \Gamma }$ is arithmetic, then Comm${\displaystyle (\Gamma )}$ is virtually isomorphic to a dense subgroup of ${\displaystyle G}$, otherwise Comm${\displaystyle (\Gamma )}$ is virtually isomorphic to ${\displaystyle \Gamma }$.

Notes

References

• MR 3753580
• Maclachlan, Colin; Reid, Alan W. (2003), The Arithmetic of Hyperbolic 3-Manifolds,
  MR 1937957
• MR 1090825