On character varieties of 3-manifold groups Misha Kapovich June 22-23, 2015
A character-buildier ◮ A dialog of a Geometric Topologist (GT) with an Algebraic Geometer (AG) ◮ GT: I would like to discuss with you character varieties of finitely-presented groups, but ... ◮ AG: Very commendable, everybody should study varieties! ◮ GT: But, let us agree, not to interrupt and not to insult each other! ◮ AG: I will try my best. ◮ GT: ... but, my knowledge of algebraic geometry is very limited at best. ◮ AG: Lazy ignoramus! ◮ GT: As I said, let us not insult each other!
What do we care about ◮ GT: The objects I care about are representation varieties Hom ( π, G ) , G = SL (2 , C ) , SU (2), π is a finitely-presented group... ◮ ... and their quotient-spaces such as Hom ( π, G ) / G , where G acts on Hom ( π, G ) by composition with inner automorphisms of G . ◮ AG (interrupting): OK, so your Hom ( π, G ) and Hom ( π, G ) / G are sets! ◮ GT: That much any lazy ignoramus knows, but I want more than that, I would like these sets to be manifolds. ◮ AG: Then you are out of luck. Do you like non-Hausdorff manifolds? ◮ GT: No, I am a geometric topologist, not a general topologist. ◮ AG: Not a problem. Hom ( π, SU (2)) / SU (2) is Hausdorff (I am assuming you mean “with classical topology”, since you dislike non-Hausdorff spaces), as for Hom ( π, SL (2 , C )) / SL (2 , C ), you just have to use its Hausdorffication.
Digression: A non-Hausdorff example ◮ Take π ∼ = Z and consider ρ : π → G = SL (2 , C ) sending the generator 1 to the matrix � 1 ◮ � 1 A = 0 1 ◮ Next, take a matrix � λ � 0 B = , λ < 1 λ − 1 0 ◮ and consider the sequence of conjugates � 1 2 λ n � A n = B n AB − n = 0 1 and corresponding conjugate representations ρ n : 1 → A n . ◮ In the limit � 1 � 0 n →∞ A n = lim , 0 1 which is not conjugate to A , of course.
Digression: A non-Hausdorff example ◮ Therefore, the projection of ρ to Hom ( π, G ) / G cannot be separated from the equivalence class of the trivial representation ρ o . ◮ Note that in this example, the orbits G · ρ and G · ρ o are distinct but their closures intersect.
What kind of a set is it? ◮ Instead of the G -orbit equivalence relation you should use the ◮ extended orbit-equivalence relation generated by ρ 1 ∼ ρ 2 ⇐ ⇒ G · ρ 1 ∩ G · ρ 2 � = ∅ . ◮ Luckily for you, since G = SL (2 , C ), this equivalence relation is the same as the orbit equivalence unless representations are conjugate to upper-triangular ones. ◮ This quotient is denoted: X ( π, G ) = Hom ( π, G ) // G . ◮ Note. The same quotient construction works for other reductive group actions on affine complex-algebraic varieties, not necessarily representation varieties. This is one of the key results of GIT, Geometric Invariant Theory . Good references are [D] and [N].
Is it a variety? ◮ GT: Wonderful! Finally, it feels like we are speaking the same language. ◮ AG: Not so fast. The best way to describe this quotient is as Spec ( R G Hom ( π, G ) ), with R = R Hom ( π, G ) the coordinate ring of the algebraic set Hom ( π, G ) ⊂ C N : � R = C [ x 1 , ..., x N ] / I Hom ( π, G ) , I Hom ( π, G ) is the ideal of Hom ( π, G ). ◮ Lastly, R G ⊂ R is the subring of G -invariants.
A commutative algebra digression ◮ Note. Here is what AG means: The trick is to think not in terms of (algebraic) sets but (polynomial) functions on these sets. Polynomial functions on an algebraic subset V ⊂ C N are the restrictions of polynomial functions on C n . The kernel of this restriction map is the ideal I V of polynomials vanishing on V . Then R = C [ x 1 , ..., x N ] / I is the coordinate ring of the variety V (the ring of polynomial functions on V ). The ring of functions on a quotient of V by G should be the ring R G of functions on V invariant under G ; we just have to find an (algebraic set) whose ring of functions is R G . Since C [ x 1 , ..., x N ] is Noetherian, its quotient R is also Noetherian and, thus, the subring R G as well (quotients and subs of Noetherian rings are Noetherian). Thus, R G is finitely generated, hence, isomorphic to a ring C [ y 1 , ..., y M ] / J . We get the variety W = { y : g ( y ) = 0 , ∀ g ∈ J } ⊂ C M , the coordinate ring of this variety is R G . Declare W to be the quotient V // G .
Is it a variety? ◮ GT: Hmm. Whatever. I liked the other description much better. Is this quotient a manifold now? ◮ AG: Alas, no. ◮ GT: But, at least, it is a variety, right? ◮ AG: It depends on what you mean by a variety . For instance it can fail to be irreducible . Most people (meaning, most algebraic geometers ) require varieties V to be irreducible, meaning that V cannot be written as a finite union of proper algebraic subsets. ◮ For instance, the algebraic subset { ( x , y ) : xy = 0 } ⊂ C 2 is reducible. The subset { x : x 2 − 1 = 0 } ⊂ C is also reducible. ◮ Convention. In what follows, an algebraic variety will be always assumed to mean an affine algebraic set, considered up to an isomorphism, no irreducibility assumption will be made.
Example of a reducible character variety ◮ π = Z / 5, X ( π/ G ) consists of 3 points, represented by ρ 1 , ρ 2 , ρ 3 , � 1 ◮ � 0 ρ 1 : 1 �→ , 0 1 ◮ � e 2 π/ 5 i � 0 ρ 2 : 1 �→ , e − 2 π/ 5 i 0 ◮ � e 4 π/ 5 i 0 � ρ 3 : 1 �→ . e − 4 π/ 5 i 0
Variety or a scheme? ◮ GT: Oh, I see, like some of us automatically assume that a manifold means a connected manifold . But I do not care about this reducibility issue, let us just call this quotient a character variety, because this is the name Culler and Shalen came up with. ◮ AG: Wait! There is also the character scheme. ◮ GT: What’s that and why do I care? Is it just another mountain to climb? ◮ AG: Because it’s there! ◮ Me (interrupting the dialog for good): Not only, as we will see, GT will indeed care about this scheme business, once we are done with the background. ◮ In fact, the “right notion” is not a scheme, but a stack, but let us not go there.
Digression: Everything as a scheme ◮ Affine algebraic schemes denoted X or X (over C ) are certain natural functors from (commutative) C -algebras to affine algebraic sets A �→ X = X ( A ), A -points of X . ◮ For us, schemes will be equivalent to their coordinate rings R = C [ x 1 , ..., x N ] / ( f 1 , ..., f m ) ◮ However, instead of only looking only at complex solutions of the system of equations f i ( x ) = 0 in N variables, ◮ we will also consider A -solutions for various algebras A , ◮ subsets of A n satisfying the system of equations f i ( a ) = 0. ◮ Example 1. A group-scheme G , e.g. SL 2 , SL 2 : C �→ SL 2 ( C ) = SL (2 , C ) .
Digression: Everything as a scheme ◮ Example 2. The scheme { x k = 0 } . R ∼ = C [ x ] / x k . ◮ If we only look at the complex solutions, the only solution we get is x = 0. ◮ But the ring C [ x ] / x k is clearly not isomorphic to the coordinate ring C = C [ x ] / ( x ) of the scheme { x = 0 } . ◮ Consider the (commutative) algebra of dual numbers C [ ǫ ] ∼ = C [ x ] / ( x 2 ) , where ǫ ↔ x ; ǫ 2 = 0. ◮ This algebra has zero divisors and, as we will see, it is a good thing.
Digression. Example: Zariski tangent bundle ◮ Take an affine algebraic scheme X given by the ideal I = ( f 1 , ..., f m ); L 1 , ..., L m are the linear parts of the polynomials f 1 , ..., f m . ◮ For the algebra A = C [ ǫ ] of dual numbers, we consider the set of A -points of X . ◮ The result, TX , is the Zariski tangent bundle of the complex variety X , the set of complex points of X . ◮ The projection ξ : TX → X is induced by the homomorphism D : a + b ǫ �→ a , where a , b ∈ C . ◮ Verification at a point, say, at 0 ∈ X : ◮ The equation f i ( z ) = 0, z = a + b ǫ ∈ A , ξ ( z ) = 0 , amounts to L i ( b ) = 0, since ǫ n = 0 , n ≥ 2 ◮ and all nonlinear terms drop out. ◮ Thus, ξ − 1 ( 0 ) ∼ = { b ∈ C n : L i ( b ) = 0 } = T 0 X .
More examples ◮ Example 3. The representation scheme R ep ( π, G ) = Hom ( π, G ) : A �→ Hom ( π, G ( A )) . ◮ Thus, complex points of this scheme are representations from π to the complex Lie group G = G ( C ). ◮ Example 4. The character scheme X ( π, G ) = R ep ( π, G ) // G . ◮ The complex points are the elements of the character variety X ( π, G ). ◮ One would like to have a similar statement over the real numbers (say, for representations to SL (2 , R ) or SU (2)), but it does not work as cleanly. ◮ One problem is that the quotient of a real algebraic set by a compact group action, in general, is not an algebraic variety, only a semialgebraic variety.
Coordinate rings of representation schemes ◮ Suppose that G is an algebraic subgroup of SL ( n , C ); for concreteness, I consider G = SL (2 , C ). ◮ Pick a generating set g 1 , . . . , g k of π and a finite set of relators r 1 , . . . , r m . ◮ Each representation ρ : π → G determines a point ( ρ ( g 1 ) , . . . , ρ ( g k )) ∈ G k ◮ which satisfies the set of equations (coming from the relators) r i (( ρ ( g 1 ) , . . . , ρ ( g k ))) − I = 0 , where I ∈ G is the identity matrix. ◮ Thus, we have N = 4 k variables x i (4 matrix entries for each direct factor of G k , G = SL (2 , C )) ◮ and M = k + 4 m equations which I write as f j ( x ) = 0, j = 1 , ..., M (this includes k equations det = 1, one per each generator). ◮ This defines the coordinate ring R = C [ x 1 , ..., x N ] / ( f 1 , ..., f M ) of our representation scheme (not variety!).
Recommend
More recommend
