In mathematics, specifically in group theory, two groups are commensurable if they differ only by a finite amount, in a precise sense.
Two groups G1 and G2 are said to be (abstractly) commensurable if there are subgroups H1 ⊂ G1 and H2 ⊂ G2 of finite index such that H1 is isomorphic to H2.
[1] For example: 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.
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.
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.
If a and b are commensurable, with smallest positive common integer multiple c, then
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. Two path-connected topological spaces are sometimes called commensurable if they have homeomorphic finite-sheeted covering spaces.
Depending on the type of space under consideration, one might want to use homotopy equivalences or diffeomorphisms instead of homeomorphisms in the definition.
By the relation between covering spaces and the fundamental group, 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.
More generally, Grigory Margulis showed that the commensurator of a lattice Γ in a semisimple Lie group G is dense in G if and only if Γ is an arithmetic subgroup of G.[6] The abstract commensurator of a group
is a connected semisimple Lie group not isomorphic to
, with trivial center and no compact factors, then by the Mostow rigidity theorem, the abstract commensurator of any irreducible lattice