Classical_Geometry-Calegari.pdf

CLASSICAL GEOMETRY — LECTURE NOTES

DANNY CALEGARI

1. A CRASH COURSE IN GROUP THEORY

A group is an algebraic object which formalizes the mathematical notion which ex-

presses the intuitive idea of symmetry . We start with an abstract deﬁnition.

Deﬁnition 1.1. A group is a set G and an operation m : GG ! G called multiplication

with the following properties:

(1) m is associative . That is, for any a;b;c 2 G ,

m(a;m(b;c)) = m(m(a;b);c)

and the product can be written unambiguously as abc .

(2) There is a unique element e 2 G called the identity with the properties that, for

any a 2 G ,

ae = ea = a

(3) For any a 2 G there is a unique element in G denoted a 1

called the inverse of a

such that

aa 1 = a 1 a = e

Given an object with some structural qualities, we can study the symmetries of that

object; namely, the set of transformations of the object to itself which preserve the structure

in question. Obviously, symmetries can be composed associatively, since the effect of a

symmetry on the object doesn’t depend on what sequence of symmetries we applied to the

object in the past. Moreover, the transformation which does nothing preserves the structure

of the object. Finally, symmetries are reversible — performing the opposite of a symmetry

is itself a symmetry. Thus, the symmetries of an object (also called the automorphisms of

an object) are an example of a group.

The power of the abstract idea of a group is that the symmetries can be studied by

themselves, without requiring them to be tied to the object they are transforming. So for

instance, the same group can act by symmetries of many different objects, or on the same

object in many different ways.

Example 1.2 . The group with only one element e and multiplication ee = e is called

the trivial group .

Example 1.3 . The integers Z with m(a;b) = a + b is a group, with identity 0 .

Example 1.4 . The positive real numbers R + with m (a;b) = ab is a group, with identity 1 .

Example 1.5 . The group with two elements even and odd and “multiplication” given by

the usual rules of addition of even and odd numbers; here even is the identity element.

This group is denoted Z=2Z .

Example 1.6 . The group of integers mod n is a group with m(a;b) = a + b mod n and

identity 0 . This group is denoted Z=nZ and also by C n , the cyclic group of length n .

DANNY CALEGARI

Deﬁnition 1.7. If G and H are groups, one can form the Cartesian product , denoted GH .

This is a group whose elements are the elements of GH where m : (GH)(GH) !

GH is deﬁned by

m((g 1 ;h 1 ); (g 2 ;h 2 )) = (m G (g 1 ;g 2 );m H (h 1 ;h 2 ))

The identity element is (e G ;e H ) .

Example 1.8 . Let S be a regular tetrahedron; label opposite pairs of edges by A;B;C .

Then the group of symmetries which preserves the labels is Z=2Z Z=2Z . It is also

known as the Klein 4 –group .

In all of the examples above, m(a;b) = m(b;a) . A group with this property is called

commutative or Abelian . Not all groups are Abelian!

Example 1.9 . Let T be an equilateral triangle with sides A;B;C opposite vertices a;b;c

in anticlockwise order. The symmetries of T are the reﬂections in the lines running from

the corners to the midpoints of opposite sides, and the rotations. There are three possible

rotations, through anticlockwise angles 0; 2=3; 4=3 which can be thought of as e;!;! 2 .

Observe that ! 1 = ! 2 . Let r a be a reﬂection through the line from the vertex a to

the midpoint of A . Then r a = r a and similarly for r b ;r c . Then ! 1 r a ! = r c but

r a ! 1 ! = r a so this group is not commutative . It is callec the dihedral group D 3 and has

6 elements.

Example 1.10 . If P is an equilateral n –gon, the symmetries are reﬂections as above and

rotations. This is called the dihedral group D n and has 2n elements. The elements are

e;!;! 2 ;:::;! n1

= ! 1

and r 1 ;r 2 ;:::;r n where r i = e for all i , r i r j = ! 2(ij)

and

! 1 r i ! = r i1 .

Example 1.11 . The symmetries of an “equilateral 1 –gon” (i.e. the unique inﬁnite 2 –valent

tree) deﬁnes a group D 1 , the inﬁnite dihedral group .

Example 1.12 . The set of 2 2 matrices whose entries are real numbers and whose de-

terminants do not vanish is a group, where multiplication is the usual multiplication of

matrices. The set of all 22 matrices is not naturally a group, since some matrices are not

invertible.

Example 1.13 . The group of permutations of the set f1 :::ng is called the symmetric group

S n . A permutation breaks the set up into subsets on which it acts by cycling the members.

For example, (3; 2; 4)(5; 1) denotes the element of S 5 which takes 1 ! 5; 2 ! 4; 3 !

2; 4 ! 3; 5 ! 1 . The group S n has n! elements. A transposition is a permutation which

interchanges exactly two elements. A permutation is even if it can be written as a product

of an even number of transpositions, and odd otherwise.

Exercise 1.14. Show that the symmetric group is not commutative for n > 2 . Identify S 3

and S 4 as groups of rigid motions of familiar objects. Show that an even permutation is

not an odd permutation, and vice versa.

Deﬁnition 1.15. A subgroup H of G is a subset such that if h 2 H then h 1 2 H , and

if h 1 ;h 2 2 H then h 1 h 2 2 H . With its inherited multiplication operation from G , H is

a group. The right cosets of H in G are the equivalence classes [g] of elements g 2 G

where the equivalence relation is given by g 1 g 2 if and only if there is an h 2 H with

g 1 = g 2 h .

Exercise 1.16. If H is ﬁnite, the number of elements of G in each equivalence class are

equal to jHj , the number of elements in H . Consequently, if jGj is ﬁnite, jHj divides jGj .

CLASSICAL GEOMETRY — LECTURE NOTES

Exercise 1.17. Show that the subset of even permutations is a subgroup of the symmetric

group, known as the alternating group and denoted A n . Identify A 5 as a group of rigid

motions of a familiar object.

Example 1.18 . Given a collection of elements fg i g G (not necessarily ﬁnite or even

countable), the subgroup generated by the g i is the subgroup whose elements are obtained

by multiplying together ﬁnitely many of the g i and their inverses in some order.

Exercise 1.19. Why are only ﬁnite multiplications allowed in deﬁning subgroups? Show

that a group in which inﬁnite multiplication makes sense is a trivial group. This fact is not

as useless as it might seem . . .

Deﬁnition 1.20. A group is cyclic if it is generated by a single element. This justiﬁes the

notation C n for Z=nZ used before.

Deﬁnition 1.21. A homomorphism between groups is a map f : G 1 ! G 2 such that

f(g 1 )f(g 2 ) = f(g 1 g 2 ) for any g 1 ;g 2 in G 1 . The kernel of a homomorphism is the sub-

group K G 1 deﬁned by K = f 1 (e) . If K = e then we say f is injective . If every

element of G 2 is in the image of f , we say it is surjective . A homomorphism which is

injective and surjective is called an isomorphism .

Example 1.22 . Every ﬁnite group G is isomorphic to a subgroup of S n where n is the

number of elements in G . For, let b : G ! f1;:::;ng be a bijection, and identify an

element g with the permutation which takes b(h) ! b(gh) for all h .

Deﬁnition 1.23. An exact sequence of groups is a (possibly terminating in either direction)

sequence

! G i ! G i+1 ! G i+2 ! :::

joined by a sequence of homomorphisms h i : G i ! G i+1 such that the image of h i is

equal to the kernel of h i+1 for each i .

Deﬁnition 1.24. If a;b 2 G , then bab 1 is called the conjugate of a by b , and aba 1 b 1

is called the commutator of a and b . Abelian groups are characterized by the property that

a conjugate of a is equal to a and every commutator is trivial.

Deﬁnition 1.25. A subgroup N G is normal , denoted N CG if for any n 2 N and

g 2 G we have gng 1 2 N . A kernel of a homomorphism is normal. Conversely, if N

is normal, we can deﬁne the quotient group G=N whose elements are equivalence classes

[g] of elements in G , and two elements g;h are equivalent iff g = hn for some n 2 N .

The multiplication is given by m([g]; [h]) = [gh] and the fact that N is normal says this is

well–deﬁned. Thus normal subgroups are exactly kernels of homomorphisms.

Example 1.26 . Any subgroup of an abelian group is normal.

Example 1.27 . Z is a normal subgroup of R . The quotient group R=Z is also called the

circle group S 1 . Can you see why?

Example 1.28 . Let D n be the dihedral group, and let C n be the subgroup generated by ! .

Then C n is normal, and D n =C n = Z=2Z .

Deﬁnition 1.29. If G is a group, the subgroup G 1 generated by the commutators in G is

called the commutator subgroup of G . Let G 2 be the subgroup generated by commutators

of elements of G with elements of G 1 . We denote G 1 = [G;G] and G 2 = [G;G 1 ] . Deﬁne

G i inductively by G i = [G;G i1 ] . The elements of G i are the elements which can be

written as products of iterated commutators of length i . If G i is trivial for some i — that

DANNY CALEGARI

is, there is some i such that every commutator of length i in G is trivial — we say G is

nilpotent .

Observe that every G i is normal, and every quotient G=G i is nilpotent.

Deﬁnition 1.30. If G is a group, let G 0 = G 1 and deﬁne G i = [G i1 ;G i1 ] . If G i is

trivial for some i , we say that G is solvable . Again, every G i is normal and every G=G i is

solvable. Obviously a nilpotent group is solvable.

Deﬁnition 1.31. An isomorphism of a group G to itself is called an automorphism . The set

of automorphisms of G is naturally a group, denoted Aut (G) . There is a homomorphism

from : G ! Aut (G) where g goes to the automorphism consisting of conjugation by

g . That is, (g)(h) = ghg 1 for any h 2 G . The automorphisms in the image of are

called inner automorphisms , and are denoted by Inn (G) . They form a normal subgroup of

Aut (G) . The quotient group is called the group of outer automorphisms and is denote by

Out (G) = Aut (G)= Inn (G) .

Deﬁnition 1.32. Suppose we have two groups G;H and a homomorphism : G !

Aut (H) . Then we can form a new group called the semi–direct product of G and H

denoted GnH whose elements are the elements of GH and multiplication is given by

m((g 1 ;h 1 ); (g 2 ;h 2 )) = (g 1 g 2 ;h 1 (g 1 )(h 2 ))

Observe that H is a normal subgroup of GnH , and there is an exact sequence

1 ! H ! GnH ! G ! 1

Example 1.33 . The dihedral group D n is equal to Z=2Zn C n where the homomorphism

: Z=2Z! Aut (C n ) takes the generator of Z=2Z to the automorphism ! ! ! 1 , where

! denotes the generator of C n .

Example 1.34 . The group Z=2Z n R where the nontrivial element of Z=2Z acts on R

by x ! x is isomorphic to the group of isometries (i.e. 1 – 1 and distance preserving

transformations) of the real line. It contains D 1 as a subgroup.

Exercise 1.35. Find an action of Z=2Z on the group S 1

so that D n is a subgroup of

Z=2ZnS 1

for every n .

Example 1.36 . The group whose elements consist of words in the alphabet a;b;A;B sub-

ject to the equivalence relation that when one of aA;Aa;bB;Bb appear in a word, they

may be removed, so for example

aBaAbb aBbb ab

A word in which none of these special subwords appears is called reduced ; it is clear that

the equivalence classes are in 1 – 1 correspondence with reduced words. Multiplication is

given by concatenation of words. The identity is the empty word, A = a 1 ;B = b 1 . In

general, the inverse of a word is obtained by reversing the order of the letters and changing

the case. This is called the free group F 2 on two generators , in this case the letters a;b .

It is easy to generalize to the free group F n on n generators , given by words in letters

a 1 ;:::;a n and their “inverse letters” A 1 ;:::;A n . One can also denote the letters A i by

the “letters” a 1

Exercise 1.37. Let G be an arbitrary group and g 1 ;g 2 :::g n a ﬁnite subset of G . Show

that there is a unique homomorphism from F n ! G sending a i ! g i .

CLASSICAL GEOMETRY — LECTURE NOTES

Example 1.38 . If we have an alphabet consisting of letters a 1 ;:::;a n and their inverses,

we can consider a collection of words in these letters r 1 ;:::;r m . If R denotes the subgroup

of F n generated by the r i and all their conjugates, then R is a normal subroup of F n and

we can form the quotient F n =R . This is denoted by

ha 1 ;:::;a n jr 1 ;:::;r m i

and an equivalent description is that it is the group whose elements are words in the a i

and their inverses modulo the equivalence relation that two words are equivalent if they

are equivalent in the free group, or if one can be obtained from the other by inserting or

deleting some r i or its inverse as a subword somewhere. The a i are the generators and

the r i the relations . Groups deﬁned this way are very important in topology. Notice that a

presentation of a group in terms of generators and relations is far from unique.

Deﬁnition 1.39. A group G is ﬁnitely generated if there is a ﬁnite subset of G which

generates G . This is equivalent to the property that there is a surjective homomorphism

from some F n to G . A group G is ﬁnitely presented if it can be expressed as hAjRi for

some ﬁnite set of generators A and relations R .

Exercise 1.40. Let G be any ﬁnite group. Show that G is ﬁnitely presented.

Exercise 1.41. Let F 2 be the free group on generators x;y . Let i : F 2 ! Z be the

homomorphism which takes x ! 1 and y ! 1 . Show that the kernel of i is not ﬁnitely

generated.

Exercise 1.42. (Harder). Let i : F 2 F 2 ! Z be the homomorphism which restricts on

either factor to i in the previous exercise. Show that the kernel of i is ﬁnitely generated but

not ﬁnitely presented.

Deﬁnition 1.43. Given groups G;H the free product of G and H , denoted GH , is the

group of words whose letters alternate between elements of G and H , with concatenation as

multiplication, and the obvious proviso that the identity is in either G or H . It is the unique

group with the universal property that there are injective homomorphisms i G : G ! GH

and i H : H ! GH , and given any other group I and homomorphisms j G : G ! I and

j H : H ! I there is a unique homomorphism c from GH to I satisfying ci G = j G

and ci H = j H .

Exercise 1.44. Show that deﬁnes an associative and commutative product on groups up

to isomorphism, and

F n = ZZZ

where we take n copies of Z in the product above.

Exercise 1.45. Show that Z=2ZZ=2Z= D 1 .

Remark 1.46 . Actually, one can extend to inﬁnite (even uncountable) products of groups

by the universal property. If one has an arbitrary set S the free group generated by S is the

free product of a collection of copies of Z , one for each element of S .

Exercise 1.47. (Hard). Every subgroup of a free group is free.

Deﬁnition 1.48. A topological group is a group which is also a space (i.e. we understand

what continuous maps of the space are) such that m : GG ! G and i : G ! G , the

multiplication and inverse maps respectively, are continuous . If G is a smooth manifold

(see appendix for deﬁnition) and the maps m and i are smooth maps, then G is called a Lie

group .

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