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Cayley Graphs

Ryan Jensen

University of Tennessee

March 26, 2014

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Group

Definition A group is a nonempty set G with a binary operation ∗ which satisfies the following: (i) closure: if a, b ∈ G, then a ∗ b ∈ G. (ii) associative: a ∗ (b ∗ c) = (a ∗ b) ∗ c for all a, b, c ∈ G. (iii) identity: there is an identity element e ∈ G so that a ∗ e = e ∗ a = a for all a ∈ G. (iv) inverse: for each a ∈ G, there is an inverse element a−1 ∈ G so that a−1 ∗ a = a ∗ a−1 = e. A group is abelian (or commutative) if a ∗ b = b ∗ a for all a, b ∈ G. We usually write ab in place of a ∗ b if the operation is known. When the group is abelian, we write a + b.

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Examples of Groups

Example: Z The integers Z = {. . . , −2, −1, 0, 1, 2, . . .} form an abelian group under the addition operation. Example: Z2 Define Z/2Z = Z2 = {¯ 0, ¯ 1}, where ¯ 0 = {z ∈ Z | z is even}, and ¯ 1 = {z ∈ Z | z is odd}. Then Z/2Z is an abelian group. Example: Zn Let n ∈ Z, and define Z/nZ = Zn = {¯ 0, ¯ 1, . . . n − 1}, where ¯ i = {z ∈ Z | remainder of z|n = i} are known as the integers modulo n. Then Z/nZ is an abelian group.

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A closer look at Z5

A multiplication (addition) table is called a Cayley Table. Let’s look at the Cayley table for the group Z5 = {0, 1, 2, 3, 4}. ∗ 1 2 3 4 1 2 3 4 1 1 2 3 4 2 2 3 4 1 3 3 4 1 2 4 4 1 2 3 Notice the table is symmetric about the diagonal, meaning the group is abelian. Also 1 generates the group, meaning that if we add 1 to itself enough times, we get the whole group.

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Other Examples of Groups

There are many examples of groups, here are a few more: Examples of Groups GL(n, R), the general linear group over the real numbers, is the group of all n × n invertible matrices with entries in R. SL(n, R), the special linear group over the real numbers, is the group of all n × n invertible matrices with entries in R whose determinant is 1. GL(2, Z13) is the group of 2 × 2 invertible matrices with entries from Z13 (as before Z13 is a group; it is actually a field since 13 is prime, but this won’t actually be needed in this presentation).

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Other Examples of Groups

Examples of Groups Sn, the symmetric group on n elements, is the group of bijections between an n element set and itself. Dn, the dihedral group of order 2n, is the group of symmetries of a regular n-gon. Many others.

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Group Isomorphisms

Definition Let H and G be groups. A function f : G → H so that f (ab) = f (a)f (b) for all a, b ∈ G is a homomorphism. If f is bijective, then f is an isomorphism. If G = H, then f is an automorphism. If there is an isomorphism between G and G, then G and H are isomorphic, written G ∼ = H. Group isomorphisms are nice since they mean two groups are the same except for the labeling of their elements.

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Subgroups

Definition A subset H of a group G is a subgroup if is itself a group under the operation of G; that H is a subgroup of G is denoted H ≤ G. Definition If Y is a subset of a group G, then the subset generated by Y is the collection of all (finite) products of elements of Y . This subgroup is denoted by Y . If Y is a finite set with elements y1, y2, . . . yn, then the notation y1, y2, . . . yn is used. A group which is generated by a single element is called cyclic.

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Examples of Subgroups

Example: Trivial Subgroups For any group G, the group consisting of only the identity is a subgroup of G, and G is a subgroup of itself. Example: Even Odd Integers A somewhat less trivial example is that the even integers are a subgroup of Z; however, the odd integers are not as there is no identity element. Example: nZ For any integer n ∈ Z, nZ = {nz|z ∈ Z} is a subgroup of Z.

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Cartesian Product

Definition Let A and B be sets. The Cartesian product of A and B is the set A × B = {(a, b) | a ∈ A, b ∈ B} Example Let A = {1, 2} and B = {a, b, c} then A × B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}

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Direct Product

Definition Given two groups G and H, their Cartesian product G × H, (denoted G ⊕ H if G and H are abelian) is a group known as the direct product (direct sum if G and H are abelian) of G and H. The group operation on G × H is done coordinate-wise. Example: Z2 ⊕ Z3 There is a group of order 6 found by taking the direct sum of Z2 and Z3, G = Z2 ⊕ Z3.

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Quotient Groups

Without going into too many technicalities about cosets, normal subgroups etc., quotient groups can be defined. Definition Let G be a group and H a normal subgroup of G. Then the quotient G/H is called the quotient group of G by H, or simply G mod H. Example: Z/nZ Z is a group, and nZ is a normal subgroup of Z. So the quotient Z/nZ is a group. (Remember Z/nZ = Zn.)

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Free Groups

Definition Let A be a set. The set A = {a1, a2, . . .} together with its formal inverses A−1 = {a−1

1 , a−1 2 , . . .} from an alphabet.

The elements of A ∪ A−1 are called letters. A word is a concatenation of letters. A reduced word is a word where no letter is adjacent to its inverse. The collection of all finite reduce words on the alphabet A is a free group on A, denote by F(A). The group operation is concatenation of words, followed by reduction if necessary.

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More Notation

Theorem Let A and B be finite sets, then F(A) is isomorphic to F(B) if and only if |A| = |B|. The above Theorem says that only the size of the alphabet is important when constructing a free group. As a result, when the alphabet is finite, i.e. |A| = n, the free group on A is denoted Fn and is called the free group of rank n, or the free group on n generators.

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Examples of Free Groups

Example: Trivial Free Group The free group on an empty generating set (or the free group

• n 0 generators) is the trivial group consisting of only the

empty word (the identity element). F(∅) = F0 = {e}. Example: F1 The free group on one generator is isomorphic to the integers. F1 = {. . . , a−2, a−1, a0 = e, a = a1, a2, . . .} F1 ∼ = Z by the map ai → i.

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Examples of Free Groups

Example: F2 F2 on generators a, b is the collection of all finite words from the letters a, b, a−1, b−1. Example elements are e = aa−1, a3, b−2, bab−1. An example of group operation: (a3) ∗ (b−2) ∗ (bab−1) = a3b−2bab−1 = a3b−1ab−1 Example: F3 F3 on generators a, b, c is done in a similar manner.

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Relators

Definition Let F be a free group. A relator on F is a word defined to be equal to the identity. For example aba−1b−1 = e is a relator in F2. The least normal subgroup N is the normal subgroup generated by a set of relators. A new group is formed by taking the quotient of F by N, F/N. Compact notation for this is S | R where S is the set of generators and R the set of relators. S | R is called a group presentation. For example {a, b} | {aba−1b−1 = e}, usually abbreviated a, b | aba−1b−1.

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Examples of Presentations

Example: Trivial Presentations | is the trivial group consisting of only the empty word. a | is F1. a, b | is F2. Example: Z2 The group given by the presentation a | a2 is isomorphic to Z2. The letter a generates F1 = {. . . , a−2, a−1, e, a, a2 . . .}. The relator a2 = e means replace a2 with e for all words in F1. The only elements left are e and a. Hence this group is isomorphic to Z2.

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A Little Graph Theory

Definition (From West) The Cartesian product of two graphs G and H, written G ⊠ H, is the graph with vertex set V (G) × V (H) specified by putting (u, v) adjacent to (u′, v′) if and only if either

1 u = u′ and vv′ ∈ E(H), or 2 v = v′ and uu′ ∈ E(G).

Definition (From West) A directed graph or digraph G is a triple consisting of a vertex set V (G), and edge set E(G), and a function assigning each edge an ordered pair of vertices. The first vertex of the ordered pair is the tail of the edge, and the second is the head; together they are the endpoints.

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Cartesian Product of Graphs

1 2 a b c (1, a) (1, b) (1, c) (2, a) (2, b) (2, c)

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Directed Graphs

Directed graphs just have directed edges.

1 2 a b c (1, a) (1, b) (1, c) (2, a) (2, b) (2, c)

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Cayley Graphs

Definition Let Γ be a group with generating set S. The Cayley Graph of Γ with respect to S, denoted ∆ = ∆(Γ; S), is the graph with V (∆) = Γ, and an edge between vertices g, h ∈ Γ if g−1h ∈ S ∪ S−1. Another way to think of the edges is if g ∈ Γ and s ∈ S ∪ S−1, then there is an edge connecting g and gs. This becomes easier to see with some examples.

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Cayley Graph of F1

We will construct a Cayley Graph for F1 ∼ = Z. Recall that F1 = a | = {. . . a−2, a−1, a0, a1, a2, . . .}. We want to draw ∆ = ∆(F1, a) = ∆(a | ). The vertices of ∆ are the elements of F1. Take any element ai ∈ F1, since a is a generator, there is an edge between ai and aia = ai+1. The result is an infinite graph, the real line R.

a−2 a−1 a0 a1 a2

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Another Cayley Graph of F1

The Cayley Graph depends both on the group and on the generating set chosen. Lets look at ∆(a, a2 | ).

a−2 a−1 a0 a1 a2 a3 a4 a5 a6

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Yet Another Cayley Graph of F1

Lets draw ∆(a2, a3 | ).

a0 a2 a3 a4 a1 a−4 a−3 a−2 a−1 a0 a1 a2 a3 a4

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Canonical Cayley Graph of F2

Lets look at ∆(a, b | ).

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Canonical Cayley Graph of F2

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Examples of how to draw Cayley Graphs

Lets draw a | a2.

a−4 a−3 a−2 a−1 a0 a1 a2 a3 a4 a0 a1

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Now lets do a | a3.

a−4 a−3 a−2 a−1 a0 a1 a2 a3 a4 a0 a1 a2

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a | a5, in a different approach.

a0 a1 a2 a3 a4 a0 a1 a2 a3 a4

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From these examples, we can see that: a | an = {a0, a1, . . . an−1}, and generated by a. Zn = {0, 1, . . . n − 1}, and generated by 1. So Zn ∼ = a | an by the map i → ai. Zn is known as the cyclic group of order n, and the Cayley graph is the cyclic graph of length n.

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∆(a, b | aba−1b−1) aba−1b−1 = e if and only if ab = ba.

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Facts about Cayley Graphs

Facts The degree of each vertex is equal to the total number of generators, i.e. |S ∪ S−1|. Relators in a group presentation correspond to cycles in the Cayley Graph. A group is abelian if and only if for each pair of generators a, b, the path aba−1b−1 is closed. The Cayley Graph of a group depends on the group, and the group presentation. A Cayley Graph exists for each finite group (each finite group has a finite presentation). Subgroups of a group can be found by looking at sub-graphs generated by elements of the group.

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Cayley Color Graphs

A Cayley Color Graph is the same as a Cayley Graph, except we no longer include the inverses of generating elements by default. Definition Let Γ be a group with generating set S. The Cayley Color Graph of Γ with respect to S, denoted ∆C = ∆C(Γ; S), is the graph with V (∆) = Γ, and an edge between vertices g, h ∈ Γ if g−1h ∈ S. So ∆C(a | ) is

a−2 a−1 a0 a1 a2

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Examples of Cayley Color Graphs

∆C(a | a5).

a0 a1 a2 a3 a4

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Examples of Cayley Color Graphs

Lets look at the Cayley Color Graph of a, b | a2, abab, b3, which is a presentation for the group D3. First notice that a2 = e means a = a−1, b3 = e means b−1 = b2. abab = e iff aba = b−1 iff aba = b2. From above, b = (aa)b(aa) = a(aba)a = ab2a.

e a b b2 ab ab2

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We can redraw the graph.

e b b2 a ab2 ab (1, a) (1, b) (1, c) (2, a) (2, b) (2, c)

Now we can compare it to a graph we have already seen, the Cartesian product of a directed path on two vertices with a directed 3 cycle.

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Theorems

Theorem Let ∆C(S | R) be a Cayley Color graph for a finite group. Then Aut (S | R) ∼ = S | R, this is not dependent on the presentation of the group. Corollary If ∆C(S1 | R1) ∼ = ∆C(S2 | R2), then S1 | R1 ∼ = S2 | R2.

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Theorems

Theorem Let H and G be finite groups with presentations PH and PG. Then there is a presentation for H × G so that ∆C(PH) ⊠ ∆C(PG) ∼ = ∆C(PH × PG). Specifically, ∆C(s1, . . . , sm | r1, . . . rt) ⊠ ∆C(sm+1, . . . , sn | rt+1, . . . rq) = ∆C(s1, . . . sn | r1, . . . , rq, sisjs−1

i

s−1

j

) for all 1 ≤ i ≤ m ≤ j ≤ n. All this Theorem is saying is that the product of groups and the product of their respective Cayley color graphs behave in a nice

• way. We won’t worry too much about the presentations.

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Examples of Cayley Color Graphs

Using the previous Theorem, we can find the standard Cayley color graph of Z2 ⊕ Z3 (Z2 = a | a2, and Z3 = a | a3). ∆C(Z2) is a directed cycle of size 2 ∆C(Z3) is a directed cycle of size 3. Hence (standard) ∆C(Z2 ⊕ Z3) = ∆C(Z2) ⊠ ∆C(Z3).

(0, 0) (0, 1) (0, 2) (1, 0) (1, 1) (1, 2)

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The Cayley of S3 = a, b | a2, b2, (ab)3 is shown below. Note that S3 is usually written S3 = {(), (12), (13), (23), (123), (132)}, the vertices are labeled this way.

() (123) (132) (12) (23) (13)

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We can now analyze the groups D3 and S3.

e b b2 a ab2 ab () (123) (132) (12) (23) (13)

Since they Cayley color graphs are isomorphic, the groups D3 and S3 are isomorphic, even though they may not have the same presentation.

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Now lets look at the groups D3 and Z2 ⊕ Z3.

e b b2 a ab2 ab (0, 0) (0, 1) (0, 2) (1, 0) (1, 1) (1, 2)

These groups are not isomorphic, as D3 is not abelian, and Z2 ⊕ Z3 is.

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Applications in Math

Theorem A subgroup of a free group is a free group. Proof (Basic Idea) Let F be a free group, and G a subgroup of F.

1 F is free of relators. 2 The Cayley graph ∆(F) is a tree (no cycles). 3 The Cayley graph ∆(G) is a connected sub-graph of ∆(F). 4 So ∆(G) is a tree. 5 So the presentation of G is free of relators. 6 Hence G is a free group.

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Applications in Math

What I use Cayley graphs for: Large Scale Geometry. Take an arbitrary space (topological, geometrical etc.). Take the Cayley graph of a group. Look at both from far away. If they look the same (quasi-isometric), then in some sense the space has the group inside it. Some interesting things about Large Scale Geometry. We are not concerned about small things. So any finite graph is trivial, as it becomes a point. So we only work with infinite graphs. Example: Any Cayley graph of a presentation of Z eventually looks like the real number line. We look at the ends of spaces, i.e. ends of a space quasi-isometric to F2.

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Applications in Computer Science

Langston et al. Application was in parallel processing. Problem was to create large graphs of given degree and diameter. Approach was to use Cayley graphs as the underlying group controls the degree, and the diameter is easy (since Cayley graphs are vertex transitive). Several records where broken for the largest graph of given degree and diameter.

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Applications in Computer Science

Here is an example group/Cayley graph from their paper. Example from Langston et al. The group was a subgroup of GL(2, Z13) consisting of all elements with determinant of ±1. The generators where 1 1

• rder 2,

11 2 8 12

• rder 52,

11 4 7 5

• rder 14.

The Cayley graph of this group has degree 5, diameter 7, and has 4368 vertices. A new record.

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References

Brian H. Bowditch, A course on geometric group theory, MSJ Memoirs, vol. 16, Mathematical Society of Japan, Tokyo, 2006. MR 2243589 (2007e:20085) Lowell Campbell, Gunnar E. Carlsson, Michael J. Dinneen, Vance Faber, Michael R. Fellows, Michael A. Langston, James W. Moore, Andrew P. Mullhaupt, and Harlan B. Sexton, Small diameter symmetric networks from linear groups, IEEE Transactions on Computers 41 (1992), no. 2, 218–220. David S Dummit and Richard M Foote, Abstract algebra, (2004), John Wiley and Sons, Inc. Thomas W Hungerford, Algebra, volume 73 of graduate texts in mathematics, Springer-Verlag, New York, 1980. Bernard Knueven, Graph automorphisms, 2014. Serge Lang, Algebra revised third edition, Springer-Verlag, 2002. James Munkres, Topology (2nd edition), 2 ed., Pearson, 2000. Piotr W Nowak and Guoliang Yu, Large scale geometry, 2012. Douglas B. West, Introduction to graph theory (2nd edition), 2 ed., Pearson, 2000. A.T. White, Graphs of groups on surfaces, volume 188: Interactions and models (north-holland mathematics studies), 1 ed., North Holland, 5 2001.