An SL_{2}(k) Character Variety Whose Dimension Jumps in Characteristic 2

Ben Martin

September 25, 2002

1 Introduction

In this note we give an example of a finitely generated group Γ such that: (I) there exists a field Ω of characteristic 2 with a discrete valuation ν_{0} and a representation ρ_{0}: Γ → SL_{2}(Ω) such that the induced action of Γ on the Bruhat-Tits tree of SL_{2}(Ω) tree is fixed-point free; (II) for every field K of characteristic zero with a discrete valuation, no homomorphism ρ: Γ → SL_{2}(K) gives rise to a fixed-point free action of Γ on the Bruhat-Tits tree of SL_{2}(K). In particular, Γ admits no ANI-representations. For terminology and background discussion, see [SZ].

If Γ were the fundamental group of a compact 3-manifold M then ρ_{0} would give an essential surface in M that is not detectable by the method of ANI representations or by looking at ideal points of curves in the character variety C(Γ, SL_{2}(ℂ)). I do not know whether such a 3-manifold exists for this particular choice of Γ, but perhaps a similar example can be constructed for appropriate M.

2 Character Varieties

Let F be a finitely generated group and let k be an algebraically closed field. We consider representations (i.e. group homomorphisms) of F into SL_{2}(k). Below, T ⊂ SL_{2}(k) denotes the group of diagonal matrices.

The set R(F, SL_{2}(k)) of representations (i.e. group homomorphisms) ρ: F → SL_{2}(k) has the structure of an affine variety over k. The group SL_{2}(k) acts on R(F, SL_{2}(k)) by conjugation: (D.ρ)(γ) := Dρ(γ)D^{−1}. Since SL_{2}(k) is a reductive algebraic group, we have a well-defined quotient variety C(F, SL_{2}(k)), the points of which correspond to Zariski-closed conjugacy classes of representations. By results of R. W. Richardson, the conjugacy class of a representation ρ is closed if and only if ρ is completely reducible. Clearly ρ is completely reducible if and only if either ρ(F) lies in some conjugate of T or ρ is irreducible. The co-ordinate ring of the affine variety C(F, SL_{2}(k)) is the ring of conjugation-invariant regular functions on R(F, SL_{2}(k)). One way to obtain such a function is as follows: given γ ∈ Γ, define Tr_{γ}: R(Γ, SL_{2}(k)) → k by Tr_{γ}(ρ) = Tr ρ(γ) = χ_{ρ}(γ), where Tr denotes trace and χ_{ρ}: Γ → k is the character of ρ. We denote by π the canonical map from R(F, SL_{2}(k)) to C(F, SL_{2}(k)).

The rest of this section is not essential for Section 3. Let S be the kalgebra generated by functions of the form Tr_{γ}, a subring of the co-ordinate ring of C(Γ, SL_{2}(k)). If S is finitely generated as a k-algebra then S is the co-ordinate ring of an affine variety X(F, SL_{2}(k)), and we have a dominant map ψ: C(F, SL_{2}(k)) → X(F, SL_{2}(k)). It is known that if k = ℂ then S is indeed finitely generated and ψ is an isomorphism (hence the name “character variety” or “variety of characters” for C(F, SL_{2}(k))). For general k it follows from results of Donkin that S is a finitely generated k-algebra and the map ψ is finite (see [M], Theorem 1.4); note that any regular conjugation-invariant function on SL_{2}(k) is a polynomial in Tr.

Conjecture 2.1 The map ψ is an isomorphism.

3 The Example

Let Γ be the group with presentation

Γ = ⟨a, b, c | a^{2} = b^{2} = c^{2} = 1, [b, cac] = [b, aca] = 1, cac = aca⟩.

Note that a, c generate a copy of the symmetric group S_{3}. We show that:

(a) if the characteristic of k is not 2 then the variety C(Γ, SL_{2}(k)) is finite, and for every ρ ∈ R(Γ, SL_{2}(k)), ρ(Γ) is finite;

(b) if k has characteristic 2 then X(Γ, SL_{2}(k)) is infinite: in particular, C(Γ, SL_{2}(k)) is infinite and there exists an affine curve in C(Γ, SL_{2}(k)) with an ideal point. (Applying the construction described in [SZ] now gives the field Ω, the discrete valuation ν_{0} and the representation ρ_{0} of (I) above.)

Because a finite group acting without inversions on a tree must fix a vertex ([S], I.6, Example 3.1), (a) implies (II) above.

Assume that the characteristic of k is not 2. Let ρ: Γ → SL_{2}(k) be any completely reducible representation. Set A = ρ(a), B = ρ(b), C = ρ(c). The hypothesis on the characteristic implies that A, B, C ∈ ±I, and part (a) follows.

Now assume that k has characteristic 2. For μ ∈ k, define ρ_{μ}: Γ → SL_{2}(k) by

ρ(a) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ), ρ(b) = ( \begin{matrix}1 & μ \\ 0 & 1 \end{matrix} ), ρ(c) = ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ).

We have

ρ(c)ρ(a)ρ(c) = ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} )( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) = ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} )( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ) = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ),

and

ρ(a)ρ(c)ρ(a) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} )( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )( \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} ) = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ),

so ρ is well-defined. Now Tr_{bc}(ρ_{μ}) = Tr (ρ_{μ}(bc)) = μ, so the function Tr_{bc} takes on infinitely many values. It follows that X(Γ, SL_{2}(k)) is infinite, as required.

References

[M] B. Martin, Reductive subgroups of reductive groups in nonzero characteristic. To appear in J. Algebra

[S] J.-P. Serre, Trees, Springer-Verlag 1980

[SZ] S. Schanuel and X. Zhang, Detection of essential surfaces with SL_{2}- trees. Math. Ann. 320 (2001), 149–165