[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2en\/wiki42\/commensurability-group-theory-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2en\/wiki42\/commensurability-group-theory-wikipedia\/","headline":"Commensurability (group theory) – Wikipedia","name":"Commensurability (group theory) – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 Equivalence relation of groups In mathematics, specifically in group theory, two groups are","datePublished":"2020-07-23","dateModified":"2020-07-23","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2en\/wiki42\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2en\/wiki42\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/64e735c526395c23dc4b8934d8cd8671d820082c","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/64e735c526395c23dc4b8934d8cd8671d820082c","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/all2en\/wiki42\/commensurability-group-theory-wikipedia\/","wordCount":3415,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Equivalence relation of groups 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.        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4 Table of ContentsCommensurability in group theory [ edit ] In topology [ edit ] The commensurator [ edit ] The abstract commensurator [ edit ] References [ edit ] Commensurability in group theory [ edit ] Two groups G first and G 2 are said to be ( abstractly ) commensurable if there are subgroups H first \u2282 G first and H 2 \u2282 G 2 of finite index such that H first is isomorphic to H 2 . [first] 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, WITH ) 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 \u0393 first and \u0393 2 of a group G are said to be commensurable if the intersection \u0393 first \u2229 C 2 is of finite index in both \u0393 first and \u0393 2 . Clearly this implies that \u0393 first and \u0393 2 are abstractly commensurable.        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Example: 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 IN are commensurable if the intersection S \u2229 T has finite codimension in both S and T . In topology [ edit ] 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 [ edit ] The commensurator of a subgroup \u0393 of a group G , denoted Comm G (\u0393), is the set of elements g of G that such that the conjugate subgroup g C g \u22121 is commensurable with \u0393. [5] In other words, CommG\u2061 ( C ) = { g \u2208 G : g C g\u22121\u2229 C \u00a0has finite index in both\u00a0C \u00a0and\u00a0g C g\u22121} . {displaystyle operatorname {Comm} _{G}(Gamma )={gin G:gGamma g^{-1}cap Gamma {text{ has finite index in both }}Gamma {text{ and }}gGamma g^{-1}}.} This is a subgroup of G that contains the normalizer N G (\u0393) (and hence contains \u0393). For example, the commensurator of the special linear group SL ( n , WITH ) in SL ( n , R ) contains SL ( n , Q ). In particular, the commensurator of SL ( n , WITH ) in SL ( n , R ) is dense in SL ( n , R ). More generally, Grigory Margulis showed that the commensurator of a lattice \u0393 in a semisimple Lie group G is dense in G if and only if \u0393 is an arithmetic subgroup of G . [6] The abstract commensurator [ edit ] The abstract commensurator of a group G {displaystyle G} , denoted Comm ( G ) {displaystyle (G)} , is the group of equivalence classes of isomorphisms \u03d5 : H \u2192 K {Displaystyle Phi: hto k} , where H {displaystyle H} and K {displaystyle K} are finite index subgroups of G {displaystyle G} , under composition. [7] Elements of Comm ( G ) {displaystyle {text{Comm}}(G)} are called commensurators of G {displaystyle G} . If G {displaystyle G} is a connected semisimple Lie group not isomorphic to PSL2( R ) {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 C \u2264 G {displaystyle Gamma leq G} is linear. Moreover, if C {displaystyle Gamma } is arithmetic, then Comm ( C ) {displaystyle (Gamma )} is virtually isomorphic to a dense subgroup of G {displaystyle G} , otherwise Comm ( C ) {displaystyle (Gamma )} is virtually isomorphic to C {displaystyle Gamma } . ^  Company & Kapovich (2018), Definition 5.13.  ^  Company & Kapovich (2018), Proposition 7.80.  ^  Company & Kapovich (2018), Corollary 8.47.  ^  Maclachlan & Reid (2003), Corollary 8.4.2.  ^  Company & Kapovich (2018), Definition 5.17.  ^  Margulis (1991), Chapter 9, Theorem B.  ^  Company & Kapovich (2018), Section 5.2.  References [ edit ] Comrade, Cornelia; Kapovich, Michael (2018), Geometric Group Theory , American Mathematical Society, ISBN\u00a0 9781470411046 , MR\u00a0 3753580 Maclachlan, Colin; Reid, Alan W. (2003), The Arithmetic of Hyperbolic 3-Manifolds , Springer Nature, ISBN\u00a0 0-387-98386-4 , MR\u00a0 1937957 Margulis, Grigory (1991), Discrete Subgroups of Semisimple Lie Groups , Springer Nature, ISBN\u00a0 3-540-12179-X , MR\u00a0 1090825       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/all2en\/wiki42\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/all2en\/wiki42\/commensurability-group-theory-wikipedia\/#breadcrumbitem","name":"Commensurability (group theory) – Wikipedia"}}]}]