Commensurability (group theory)

In mathematics, specifically in group theory, two groups are commensurable if they differ only by a finite amount, in a precise sense. The commensurator of a subgroup is another subgroup, related to the normalizer.
Commensurability in group theory.
Two groups "G"1 and "G"2 are said to be (abstractly) commensurable if there are subgroups "H"1 ⊂ "G"1 and "H"2 ⊂ "G"2 of finite index such that "H"1 is isomorphic to "H"2. For example:
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. 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 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.
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 "g"Γ"g"−1 is commensurable with Γ. In other words,
This is a subgroup of "G" that contains the normalizer N"G"(Γ) (and hence contains Γ).
For example, the commensurator of the special linear group "SL"("n",Z) in "SL"("n",R) contains "SL"("n",Q). In particular, the commensurator of "SL"("n",Z) in "SL"("n",R) is dense in "SL"("n",R). 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".
The abstract commensurator.
The abstract commensurator of a group formula_2, denoted Commformula_3, is the group of equivalence classes of isomorphisms formula_4, where formula_5 and formula_6 are finite index subgroups of formula_2, under composition. Elements of formula_8 are called commensurators of formula_2.
If formula_2 is a connected semisimple Lie group not isomorphic to formula_11, with trivial center and no compact factors, then by the Mostow rigidity theorem, the abstract commensurator of any irreducible lattice formula_12 is linear. Moreover, if formula_13 is arithmetic, then Commformula_14 is virtually isomorphic to a dense subgroup of formula_2, otherwise Commformula_14 is virtually isomorphic to formula_13.