And again, equivalence sub f immediately inherits the properties of equality, which makes it an equivalence relation. Commensurable definition of commensurable by merriamwebster. Then r is an equivalence relation and the equivalence classes of r are the sets of f. The equivalence class of x is the set of all elements in x which get mapped to fx, i. Wikipedia 2009, for example, defines this mathematical usage as follows. To every pair of an ergodic discrete equivalence relation r and an equivalence subrelation s, they associated a very special cocycle.
The notion of commensurability in group theory and geometry. Its noncommutative algebra of coordinates is given by the algebra ak c gk obtained by taking the quotient by scaling gk rkc of the groupoid of the equivalence relation of commensurability. Solution 3, 1 is the single ordered pair which needs to be added to r to make it the smallest equivalence relation. The hypotenuse of an isosceles right triangle is incommensurable with its side. The case of commensurability classes of 1dimensional klattices up to scaling is more delicate. Examples of groups that are measure equivalent to the free group. An equivalence relation is a relationship on a set, generally denoted by. We say that a collection m of hyperbolic 3manifolds is commensurably in. C and a word in the generators of this presentation, and decides whether or not this word represents the. We introduce a new relation characteristic equivalence on the class of arithmetic lattices, stronger than weak commensurability. Commensurability as a concept originates in the field of geometry, meaning of common measure. A, let a x be the set of all elements of a that are equivalent to x. Equivalence relation and partitions an equivalence relation on a set xis a relation which is re. Commensurability for hyperbolic groups article pdf available in transactions of the american mathematical society 3715 january 2017 with 12 reads.
A relation r on a set a is an equivalence relation if and only if r is re. The ordered pair part comes in because the relation ris the set of all x. Then the equivalence classes of r form a partition of a. The commensurability relation for finitely generated. Then is an equivalence relation with equivalence classes 0evens, and 1odds. K commensurable subgroups have the same commensurator, while weakly commensurable subgroups have conjugate commensurators. There does not exist a borel way of selecting an isomorphism class within each commensurability class of finitely generated groups. This equivalence relation is known as the kernel of f. Since f is a partition, for each x in s there is one and only one set of f which contains x. An equivalence relation on a set s, is a relation on s which is.
There are three main observations which make up the proof of theorem 4. Regular expressions 1 equivalence relation and partitions. We observe that if we use the notion of representation equivalence of lattices, then schanuels conjecture can be avoided. A relation that is all three of reflexive, symmetric, and transitive, is called an equivalence relation. Equivalence relations a binary relation is an equivalence relation iff it has these 3 properties. It is of course enormously important, but is not a very interesting example, since no two distinct objects are related by equality. Isomorphism versus commensurability for a class of. Weak commensurability is also an equivalence relation.
The notion of commensurability in group theory and geometry luisa paoluzzi latp marseilles france. If two quantities can be measured in the same units, they are commensurable. Commensurability definition of commensurability by the. Two groups g 1 and g 2 are said to be abstractly commensurable if there are subgroups h 1. Example 3 let r be the equivalence relation in the set z of integers given by r a, b. Finite volume hyperbolic 3manifolds whose fundamental. G 2 of finite index such that h 1 is isomorphic to h 2. The set of real numbers r can be partitioned into the set of.
Reflexive xx symmetric if xy then yx transitive if xy and yz then xz rst note. We illustrate how to show a relation is an equivalence relation or how to show it is not an equivalence. The simplest instances of me groups are commensurable groups or more. Moreover, it also turns out to be equivalent to the condition that s is the kernel of an almost. An equivalence is just the relation e induced by a partition is any relation that satisfies 3 conditions. We show that the isomorphism relation for finitely generated solvable groups of class 3 is a weakly universal countable borel equivalence relation. This video is part of a discrete math course taught by dr. Mathematics exactly divisible by the same unit an integral number of times. Pdf commensurability and representation equivalent. Ifp0is a copy of pobtained by displacing palong the perpendicular to pthrough its center, then p0and pcan be joined to obtain a regular hyperbolic drum. And the theorem that we have is that every relation r on a set a is an equivalence relation if and only if it in fact is equal to equivalence sub f for some function f. We introduce a new relation characteristic equivalence on the class of arith metic lattices, stronger than weak commensurability. On the normalizing groupoids and the commensurability.
The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest. A partition of a set x is a set p fc i x ji 2ig such that i2i c i x covering property 8i 6 s c. Show that the equivalence class of x with respect to p is a, that is that x p a. A group is finite if and only if it is commensurable with the trivial group. Commensurability and representation equivalent arithmetic lattices. The commensurability relation for finitely generated groups the commensurability relation for finitely generated groups thomas, simon 20091101 00. An equivalence relation on a set xis a relation which is re. An equivalence relation defines how we can cut up our pie how we partition our set of values into slices. The equivalence relation of commensurability on the space of qlattices is subtle enough an operation that the resulting quotient can only be described e ciently through noncommutative geometry it is crucial for this that one does not restrict to the invertible ones. The cartesian product a bof sets aand bis the set of all possible ordered pairs with the rst element drawn from aand the second from b. Write the ordered pairs to be added to r to make it the smallest equivalence relation.
Regular equivalence captures role concept really well two actors are equivalent if they have the same. For part a, i proved it to be true by showing that its reflexive, symmetric and transitive. Equivalence relations r a is an equivalence iff r is. That is, any two equivalence classes of an equivalence relation are either mutually disjoint or identical. An equivalence relation is a relation which looks like ordinary equality of numbers, but which may hold between other kinds of objects. Kuhn introduces the term to talk about scientific paradigms. Pdf this paper proposes a reading of the history of equivalence in. Further, the results are applicable in a sarithmetic setting. The quotient of x by, denoted x and called x mod, is the set of equivalence classes for the.
Any two finitely generated free groups on at least 2 generators are commensurable with each other. Define a relation on s by x r y iff there is a set in f which contains both x and y. Read and learn for free about the following article. In the meantime, in 7, feldman, sutherland and zimmer developed the theory of inclusions of equivalence relations. Equality on any set x y iff x y over the set of strngs a,b,c. Gromovs invariant and the volume of a hyperbolic manifold let pbe a regularkgon in h3 with all vertices on s2 1. A relation ris a subset of x x, but equivalence relations say something about elements of x, not ordered pairs of elements of x. Equivalence relation definition of equivalence relation. More generally, a function may map equivalent arguments under an equivalence relation x on x to equivalent values under an equivalence relation y on y.
Equivalence relation definition is a relation such as equality between elements of a set such as the real numbers that is symmetric, reflexive, and transitive and. Equivalence relations are a way to break up a set x into a union of disjoint subsets. Commensurability and symmetry are thus two distinct mathematical no. Conversely, given a partition fa i ji 2igof the set a, there is an equivalence relation r that has the sets a. Having the book only as a pdf image, i do not know how these larger leaves were. Equivalence relations now we group properties of relations together to define new types of important relations. Interestingly, birkhoff defined an equivalence relation to be any. Commensurable definition is having a common measure. Here are three familiar properties of equality of real numbers. Having the book only as a pdf image, i do not know how these larger.