Commensurability (mathematics)

In mathematics , the term commensurability is used in various contexts, in addition to its classic use (see incommensurability (mathematics) ), for example, in group theory and topology .

Classic use of the term commensurability

Two real numbers a and b are called commensurable ( Latin: measurable together ) if they are integral multiples of a suitable third real number c .

This condition is (for ) equivalent to the fact that the ratio of and is a rational number : ${\ displaystyle b \ not = 0}$ ${\ displaystyle x}$ ${\ displaystyle a}$ ${\ displaystyle b}$

{\ displaystyle {\ frac {a} {b}} = x \ in \ mathbb {Q}}

.

(In addition, every real number is commensurate.) ${\ displaystyle b = 0}$

For example, all rational numbers are commensurable to one another. The side of a square and the length of its diagonal are incommensurable because, according to the Pythagorean Theorem, is , and the assumption that this is a fraction, can be refuted. On the other hand, and are commensurate with one another. ${\ displaystyle a}$ ${\ displaystyle d}$ ${\ displaystyle {\ tfrac {d} {a}} = {\ sqrt {2}}}$ ${\ displaystyle {\ sqrt {2}}}$ ${\ displaystyle 2 {\ sqrt {2}}}$

Group theory

Subgroups of a given group

Be a given group, then called two subgroups each other commensurable if the average of finite index in both than does. ${\ displaystyle G}$ ${\ displaystyle H_ {1}, H_ {2} \ subset G}$ ${\ displaystyle H_ {1} \ cap H_ {2}}$ ${\ displaystyle H_ {1}}$ ${\ displaystyle H_ {2}}$

For example, all subgroups of the group of integers are commensurable to one another: all subgroups of are of the form for suitable , the average has finite index or in or . ${\ displaystyle G = \ mathbb {Z}}$ ${\ displaystyle H_ {1} = m \ mathbb {Z}, H_ {2} = n \ mathbb {Z}}$ ${\ displaystyle m, n \ in \ mathbb {N}}$ ${\ displaystyle H_ {1} \ cap H_ {2} = \ operatorname {kgV} (m, n) \ mathbb {Z}}$ ${\ displaystyle {\ frac {\ operatorname {kgV} (m, n)} {n}}}$ ${\ displaystyle {\ frac {\ operatorname {kgV} (m, n)} {m}}}$ ${\ displaystyle H_ {1}}$ ${\ displaystyle H_ {2}}$

Abstract groups

Two groups are called commensurable to one another if there is an isomorphism of two subgroups of finite index. ${\ displaystyle H_ {1}, H_ {2}}$ ${\ displaystyle \ phi \ colon K_ {1} \ rightarrow K_ {2}}$ ${\ displaystyle K_ {1} \ subset H_ {1}, K_ {2} \ subset H_ {2}}$

This is especially the case for commensurable subgroups of a given group, here you can use and . ${\ displaystyle K_ {1} = K_ {2} = H_ {1} \ cap H_ {2}}$ ${\ displaystyle \ phi = id}$

Two subgroups of a given group that are commensurable as abstract groups need not necessarily be commensurable subgroups in the sense of the previous section.

Geometric group theory

On finitely generated groups, the word metric can be used to define a structure of a metric space on the Cayley graph . Commensurable groups have quasi-isometric Cayley graphs; the reverse is generally not true. But there are a number of special cases in which the reverse also applies. For example, a group is quasi-isometric to if and only if it is (abstractly) commensurable to ; or it is quasi-isometric to a free group if and only if it is commensurable to the free group (abstractly). If there are fundamental groups of two non-compact hyperbolic manifolds of finite volume and of the same dimension , then they are quasi-isometric if and only if they are commensurable to one another (as subgroups of the isometric group of hyperbolic space). In contrast, all fundamental groups of compact hyperbolic manifolds of a given dimension are quasi-isometric to one another, but they are not always commensurable to one another. ${\ displaystyle \ mathbb {Z} ^ {n}}$ ${\ displaystyle \ mathbb {Z} ^ {n}}$ ${\ displaystyle \ Gamma _ {1}, \ Gamma _ {2}}$ ${\ displaystyle n \ geq 3}$

topology

Two topological spaces are called commensurable if there is a topological space with finite overlays . ${\ displaystyle X_ {1}, X_ {2}}$ ${\ displaystyle X}$ ${\ displaystyle p_ {1} \ colon X \ rightarrow X_ {1}, p_ {2} \ colon X \ rightarrow X_ {2}}$

For example, different lens spaces are commensurate to one another because the sphere finitely overlays them all .

The topological and group-theoretical concepts of commensurability are related as follows. When two topological spaces are commensurable with each other, then their are fundamental groups commensurable since and each contain a subgroup of finite index, to be isomorphic. ${\ displaystyle \ pi _ {1} X_ {1}}$ ${\ displaystyle \ pi _ {1} X_ {2}}$ ${\ displaystyle \ pi _ {1} X}$

For spaces that have a common universal superposition , for example hyperbolic manifolds of a given dimension, the converse also applies: two such spaces are commensurable if and only if their fundamental groups as subgroups of the group of homeomorphisms are commensurable. ${\ displaystyle {\ widetilde {X}}}$ ${\ displaystyle {\ widetilde {X}}}$

Web links

Luisa Paoluzzi: The notion of commensurability in group theory and geometry

Individual evidence

↑ Richard Evan Schwartz : The quasi-isometry classification of rank one lattices. Inst. Hautes Études Sci. Publ. Math. No. 82: 133-168 (1996) (1995).

[1] Richard Evan Schwartz : The quasi-isometry classification of rank one lattices. Inst. Hautes Études Sci. Publ. Math. No. 82: 133-168 (1996) (1995).