Commensurability (group theory)

Last updated April 05, 2019

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.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is usually denoted H ≤ G, read as "H is a subgroup of G".

Commensurability in group theory

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 mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

In mathematics, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.

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.

In mathematics, the free groupF_S over a given set S consists of all expressions that can be built from members of S, considering two expressions different unless their equality follows from the group axioms. The members of S are called generators of F_S. An arbitrary group G is called free if it is isomorphic to F_S for some subset S of G, that is, if there is a subset S of G such that every element of G can be written in one and only one way as a product of finitely many elements of S and their inverses.

In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant. The matrices A and -A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic.

In mathematics, genus has a few different, but closely related, meanings. The most common concept, the genus of an (orientable) surface, is the number of "holes" it has. This is made more precise below.

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.

Intersection (set theory) concept in mathematics specific to the field of set theory

In mathematics, the intersectionA ∩ B of two sets A and B is the set that contains all elements of A that also belong to B, but no other elements.

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 mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the transcendental numbers, such as $π$ (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.

Generating set of a group Subset of a group such that all group elements can be expressed by finitely many group operations on its elements

In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their inverses.

In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio a/b is a rational number; otherwise a and b are called incommensurable. There is a more general notion of commensurability in group theory.

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.

Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act.

In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination of finitely many elements of the finite set S and of inverses of such elements.

In mathematics, a metric space is a set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

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.^[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,

\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 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.^[6]

The abstract commensurator

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

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

Notes

↑ Druțu & Kapovich (2018), Definition 5.13.
↑ Druțu & Kapovich (2018), Proposition 7.80.
↑ Druțu & Kapovich (2018), Corollary 8.47.
↑ Maclachlan & Reid (2003), Corollary 8.4.2.
↑ Druțu & Kapovich (2018), Definition 5.17.
↑ Margulis (1991), Chapter IX, Theorem B.
↑ Druțu & Kapovich (2018), Section 5.2.

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References

Druțu, Cornelia; Kapovich, Michael (2018), Geometric Group Theory, American Mathematical Society, ISBN 9781470411046, MR 3753580
Maclachlan, Colin; Reid, Alan W. (2003), The Arithmetic of Hyperbolic 3-Manifolds, Springer Nature, ISBN 0-387-98386-4, MR 1937957
Margulis, Grigory (1991), Discrete Subgroups of Semisimple Lie Groups, Springer Nature, ISBN 3-540-12179-X, MR 1090825

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[1] Druțu & Kapovich (2018), Definition 5.13.

[2] Druțu & Kapovich (2018), Proposition 7.80.

[3] Druțu & Kapovich (2018), Corollary 8.47.

[4] Maclachlan & Reid (2003), Corollary 8.4.2.

[5] Druțu & Kapovich (2018), Definition 5.17.

[6] Margulis (1991), Chapter IX, Theorem B.

[7] Druțu & Kapovich (2018), Section 5.2.