AN ERGODIC ACTION OF THE OUTER AUTOMORPHISM GROUP OF A FREE GROUP

WILLIAM M. GOLDMAN

Abstract. For n > 2, the action of the outer automorphism group of the rank n free group F_{n} on Hom(F_{n}, SU(2))/SU(2) is ergodic with respect to the Lebesgue measure class.

Introduction

Let F_{n} be a free group of rank n > 1 and let G be a compact Lie group. Then Hom(F_{n}, G) admits a natural volume form which is invariant under Aut(F_{n}). This volume form descends to a finite measure on the character variety Hom(F_{n}, G)/G which is invariant under Out(F_{n}). The purpose of this note is to prove:

Theorem. Suppose that G is a connected group locally isomorphic to a product of copies of SU(2) and U(1). If n > 2, then the Out(F_{n})-action on Hom(F_{n}, G)/G is ergodic.

We conjecture that Out(F_{n}) is ergodic on each connected component of Hom(F_{n}, G)/G for every compact Lie group G and n > 2.

When G = U(1), then this action is just the action of GL(n, ℤ) on the n-torus ℝ^{n}/ℤ^{n}, which is well known to be ergodic. In fact, certain cyclic subgroups of GL(n, ℤ) act ergodicly.

The proof relies heavily on [2], both in its outline of proof and a key result. When n = 2, the action is not ergodic, since it preserves the function

Hom(F_{n}, G)/G ⟶^{κ} [−2, 2]
[ρ] ⟼ tr([X_{1}, X_{2}])

Date: June 20, 2005.

1991 Mathematics Subject Classification. 57M05 (Low-dimensional topology), 22D40 (Ergodic theory on groups).

Key words and phrases. character variety, free group, outer automorphism group, Nielsen transformation, ergodic equivalence relation.

The author gratefully acknowledges support from National Science Foundation grants DMS-0405605 and DMS-0103889 .

where X_{1}, X_{2} are a pair of free generators for F_{2}. However, for each −2 ≤ t ≤ 2, the action is ergodic on κ^{−1}(t).

When π is the fundamental group of a closed surface, then Pickrell and Xia [4] have proved Out(π) is ergodic on Hom(π, G)/G for any compact Lie group G.

As in [2], the methods here apply when G is any Lie group having simple factors U(1) and SU(2). In particular, since this class of groups is closed under the operation of taking direct products, the action of Out(F_{n}) is ergodic on

Hom(F_{n}, G × G)/(G × G) ⟷ Hom(F_{n}, G)/G × Hom(F_{n}, G)/G.

As in [2], the action of Out(F_{n}) on Hom(F_{n}, G)/G is weak-mixing, that is:

Corollary. The only invariant finite-dimensional subrepresentation of the induced unitary representation of Out(F_{n}) on L^{2}(Hom(F_{n}, G)/G) consists of constants.

1. Ergodic theory of the SU(2)-character variety

Let {X_{1}, …, X_{n}} be a set of free generators for F_{n} and let

X_{0} = X_{n}^{−1} … X_{1}^{−1}.

Then F_{n} is the fundamental group of an n+1-holed sphere S_{n+1}, where the X_{0}, X_{1}, …, X_{n} correspond to components of ∂S_{n+1}. The mapping class group Γ_{n+1} of S_{n+1} embeds in Out(F_{n}) as the subgroup preserving the conjugacy classes of the cyclic subgroups ⟨X_{i}⟩ for i = 0, …, n.

The proof proceeds as follows. Let Hom(F_{n}, G)/G ⟶^{f} ℝ be an Out(F_{n})-invariant measurable function. We show that f is constant almost everywhere.

The main result of [2] applied to the surface S_{n+1} gives the following:

Proposition 1.1. The mapping

Hom(F_{n}, G)/G ⟶^{t_{∂}} [−2, 2]^{n+1}

[ρ] ⟼ [tr(ρ(X_{0})) ; ⋮ ; tr(ρ(X_{n}))]

is an ergodic decomposition for the action of Γ_{n+1}. That is, for every Γ_{n+1}-invariant measurable function

Hom(F_{n}, G)/G ⟶^{h} ℝ

there exists a measurable function

[−2, 2]^{n+1} ⟶^{H} ℝ

such that h = H ∘ t_{∂} almost everywhere.

Using the embedding of the mapping class group

Γ_{n+1} ↪ Out(F_{n})

as above, the Out(F_{n})-invariant function f is Γ_{n+1}-invariant, and hence factors through t_{∂}. Denote the corresponding function on [−2, 2]^{n+1} by

(1.1)  [−2, 2]^{n+1} ⟶^{F} ℝ.

2. The case of rank n = 3

First consider the case n = 3. Following the notation of [1, 2], denote the generators by

A = X_{1}, B = X_{2}, C = X_{3}, D = X_{0}

so that A, B, C, D are subject to the relation

(2.1)  ABCD = 1.

A representation ρ is determined by its values

(ρ(A), ρ(B), ρ(C)) ∈ G^{3}

on the generators A, B, C and

ρ(D) = ρ(C)^{−1}ρ(B)^{−1}ρ(A)^{−1}.

For brevity we omit writing ρ, just referring to, for example, ρ(A) as just A when the context is clear.

2.1. Trace coordinates. The equivalence class [ρ] is determined by the seven functions

a = tr(A)
b = tr(B)
c = tr(C)
d = tr(D) = tr(D^{−1}) = tr(ABC)
x = tr(AB)
y = tr(BC)
z = tr(CA)

subject to the polynomial relation

(2.2)  x^{2}+y^{2} + z^{2} + xyz =
(ab + cd)x + (ad + bc)y + (ac + bd)z
+ (4 − a^{2} − b^{2} − c^{2} − d^{2} − abcd).

In other words, the SL(2, ℂ)-character variety of F_{3} is the hypersurface in ℂ^{7} defined by (2.2).

When a, b, c, d ∈ ℝ the topology of the set of ℝ-points is analyzed in [1]. In particular the SU(2)-character variety is the union over the set V of all

(a, b, c, d) ∈ [−2, 2]^{4}

satisfying

0 ≥ Δ(a, b, c, d) = (2(a^{2} + b^{2} + c^{2} + d^{2}) − abcd − 16)^{2}
− (4 − a^{2})(4 − b^{2})(4 − c^{2})(4 − d^{2})

of compact components of the cubic surface in ℝ^{3} satisfying (2.2). Here is an alternate description with which it is easier to work.

2.2. Rank two free groups. The SU(2)-character variety of F_{2} is the subset V_{3} ⊂ ℝ^{3} defined by traces

(x_{1}, x_{2}, x_{3}) ∈ [−2, 2]^{3}

satisfying the inequality

(2.3)  x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{1}x_{2}x_{3} ≤ 4

which is depicted in Figure 1.

A quadruple (a, b, c, d) ∈ [−2, 2]^{4} is the image of an SU(2)-character if and only if there exists y ∈ ℝ such that both triples (a, d, y) and (b, c, y) lie in V_{3}.

Using (2.2), we determine the condition that (a, d, y) ∈ V_{3}. Apply (2.2) to x_{1} = y, x_{2} = a, x_{3} = d to see that (a, d, y) ∈ V_{3} if and only if y lies in the interval

Y (a, d) := [y_{−}(a, d), y_{+}(a, d)]

with endpoints

y_{±}(a, d) := \frac{ad ± \sqrt{(4 − a^{2})(4 − d^{2})}}{2}.

For any (a, d) ∈ [−2, 2]^{2}, the interval Y (a, d) is nonempty, so that the restriction of the projection

V_{3} ⟶^{Π_{a,d}} [−2, 2]^{2}

[\begin{matrix}a \\ b \\ c \\ d \\ x \\ y \\ z \end{matrix}] ⟼ [\begin{matrix}a \\ d \end{matrix}]

is onto. Furthermore the fiber V_{3}(a, d) of the surjection

t_{∂}(V_{3}) ↠^{Π_{a,d}} [−2, 2]^{2}

[\begin{matrix}a \\ b \\ c \\ d \end{matrix}] ⟼ [\begin{matrix}a \\ d \end{matrix}]

consists of all (b, c) ∈ [−2, 2]^{2} such that

Y (b, c) ∩ Y (a, d) ≠ ∅.

If −2 < y < 2, then the set of (b, c) such that y ∈ Y (b, c) is the closed elliptical region Ē_{y} inscribed in the square [−2, 2]^{2} at the four points

(2, y), (y, 2), (−2, −y), (−y, 2),

depicted in Figure 3. If y = ±2, then the set of (b, c) such that y ∈ Y (b, c) is the line segment

{(b, ∓b) | −2 ≤ b ≤ 2}.

For fixed a, d ∈ [−2, 2], the fiber Π^{−1}(V_{3}(a, d)) equals

⋃_{y∈Y (a,d)} {(b, c) | y ∈ Y (b, c)},

depicted in Figure 4.

Figure 1. The SU(2)-character variety of a rank two free group is the region inside this surface. The surface, a rounded tetrahedron, consists of characters of abelian representations. It is a quotient of the 2-torus by an involution, where the fixed points of the involution correspond to the vertices of the tetrahedron.

Figure 2. The SU(2)-character variety has three foliations by ellipses, corresponding to the three coordinate planes. This figure depicts projections of the leaves of one of the foliations into a coordinate plane. Leaves of the other two foliations project to to horizontal and vertical line segments, respectively.

Figure 3. For y_{0} = −1.2, the set of possible (b, c) ∈ [−2, 2]^{2} for which (b, c, y_{0}) is the character of an SU(2)- representation is the interior of the ellipse E_{1.2} inscribed in b[−2, 2]^{2} at the four points (±2, ∓1.2) and (∓1.2, ±2).

Figure 4. Here are the ellipses drawn for the interval Y = [0, 1.8]. The union of the closed elliptical regions is the (b, c)-projection of the V_{4}(a, d) where ad = y_{+}y_{+} = 1.8 and a^{2} + d^{2} = (since y_{−} = 0), for example a = 1.69, d = 1.062.

2.3. A non-geometric automorphism. The automorphism α ∈ Aut(F_{3}) defined by:

A ⟼^{α} A
B ⟼^{α} BA^{−1}
C ⟼^{α} AC
D ⟼^{α} D

induces the following automorphism of the character variety:

[\begin{matrix}a \\ b \\ c \\ d \\ x \\ y \\ z \end{matrix}] ⟼^{α^{∗}} [\begin{matrix}a \\ x \\ ac − z \\ d \\ ax − b \\ y \\ c \end{matrix}]

Since α^{∗} preserves the coordinates a, d, y, this automorphism restricts to a diffeomorphism on the level sets a = a_{0}, d = d_{0} and y = y_{0} and the restriction is linear:

[\begin{matrix}x \\ b \\ c \\ z \end{matrix}] ⟼^{α^{∗}} [\begin{matrix}a_{0} & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & a_{0} & −1 \\ 0 & 0 & 1 & 0 \end{matrix}] [\begin{matrix}x \\ b \\ c \\ z \end{matrix}]

The following elementary fact (whose proof is omitted) is useful:

Lemma 2.1. Let −2 < a < 2. The linear transformation

ℝ^{2} ⟶^{L_{a}} ℝ^{2}
[\begin{matrix}x \\ y \end{matrix}] ⟼ [\begin{matrix}ax − y \\ x \end{matrix}]

preserves the positive definite quadratic form

[\begin{matrix}x \\ y \end{matrix}] ⟼^{Q} x^{2} − axy + y^{2}

and lies in the linear flow generated by the vector field

Υ := (\frac{a}{2}x − y)∂_{x} + (x − \frac{a}{2}y)∂_{y}.

The trajectories of Υ are the level sets of Q, which are ellipses and L_{a}.acts by rotation on these ellipses by angle

(2.4)  θ = 2 cos^{−1}(a/2).

If θ ∉ πℚ, then L_{a} has infinite order and is ergodic on each Q-level set.

The 4-dimensional affine subspaces of ℝ^{7} corresponding to levels of (a_{0}, d_{0}, y_{0}) split as products of two affine 2-planes (corresponding to levels of (x, b) and (c, z) respectively). Evidently the linear map α^{∗} on these 4-planes splits as a direct sum of two copies of the linear map L_{a_{0}}. It preserves the trajectories of the linear vector field

A = (\frac{a_{0}}{2} x − b)∂_{x} + (x − \frac{a_{0}}{2} b)∂_{b}

+ (\frac{a_{0}}{2} c − z)∂_{c} + (c − \frac{a_{0}}{2} z)∂_{z}.

The zeroes of this vector field consist of the origin

[\begin{matrix}x \\ b \\ c \\ z \end{matrix}] = [\begin{matrix}0 \\ 0 \\ 0 \\ 0 \end{matrix}]

and, when a_{0} = ±2, the squares defined by

x = ±b, z = ±c.

All other trajectories are ellipses. The transformation α^{∗} acts by rotation along these ellipses through angle θ given by (2.4). When θ/π is irrational, this action is ergodic. Thus, for almost every a_{0} ∈ [−2, 2], the restriction of α^{∗} to these ellipses is ergodic. On a set of full measure, the function f is constant along the projections of trajectories of A.

Fix (a_{0}, d_{0}) ∈ (−2, 2)^{2} and consider the equivalence relation ∼ on V_{3}(a_{0}, d_{0}) generated by the projections of trajectories of A.

Lemma 2.2. For a_{0}, d_{0} ≠ ±2, all points in V_{3}(a_{0}, d_{0}) are ∼-equivalent.

Proof. Since V_{3}(a_{0}, d_{0}) is connected, it suffices to prove that each equivalence class is open. To this end, suppose that (b_{0}, c_{0}) ∈ V_{3}(a_{0}, d_{0}); we show that every (b, c) sufficiently close to (b_{0}, c_{0}) is equivalent to (b_{0}, c_{0}).

The image of the tangent vector A at (x, b, c, z) under the differential of the coordinate projection

[\begin{matrix}x \\ b \\ c \\ z \end{matrix}] ⟼^{Π_{b,c}} [\begin{matrix}b \\ c \end{matrix}]

is

(Π_{b,c})_{∗}A = (x − \frac{a_{0}}{2} b)∂_{b} + (\frac{a_{0}}{2} c − z)∂_{c}

The fiber Π_{b,c}^{−1}(b_{0}, c_{0}) is an interval. For some (and hence almost every) (x, z) ∈ Π_{b,c}^{−1}(b_{0}, c_{0}), the vector

(Π_{b,c})_{∗}A (b_{0}, x, z, c_{0})

is nonzero. Choose such an (x_{0}, z_{0}). For any open neighborhood U of (x_{0}, z_{0}), the values (Π_{b,c})_{∗}A (b_{0}, x, z, c_{0}) for (x, z) ∈ U, span ℝ^{2}.

Let Φ_{t} denote the flow generated by A and choose an open neighborhood U_{0} of (b_{0}, x_{0}, z_{0}, c_{0}) in V_{3}(a_{0}, d_{0}). The differential of the map

ℝ × ({b_{0}} × U_{0} × {c_{0}}) ⟶ ℝ^{2}
(t, b_{0}, x, z, c_{0}) ⟼ Π_{b,c}(Φ_{t}(b_{0}, x, z, c_{0}))

at (0, b_{0}, x_{0}, z_{0}, c_{0}) is onto. The inverse function theorem guarantees an open neighborhood of (0, b_{0}, x_{0}, z_{0}, c_{0}) mapping onto an open neighborhood of (b_{0}, c_{0}), as desired. □

Thus, for almost every (a_{0}, d_{0}) ∈ [−2, 2]^{2}, the function F of (1.1) is constant along the level surfaces Π_{a,d}^{−1}(a_{0}, d_{0}), and hence factors through the projection Π_{a,d}:

F (a, b, c, d) = F (a, d).

Applying the same argument to the automorphism

A ⟼^{γ} CA
B ⟼^{γ} B
C ⟼^{γ} C
D ⟼^{γ} DC^{−1}

implies that F factors through the projection Π_{(b,c)} and F is almost everywhere constant.

Thus the function f, which is invariant under the mapping class group Γ_{4} and the automorphisms α_{∗}, γ_{∗}, must be constant almost everywhere. This completes the proof of the Theorem when n = 3.

3. General rank

The case of rank n > 3 follows easily from the case n = 3. The following elementary lemma is useful:

Lemma 3.1. Let G be a compact Lie group. Then Out(F_{n}) is ergodic on Hom(F_{n}, G)/G if and only if Aut(F_{n}) is ergodic on Hom(F_{n}, G). More generally, if Γ ⊂ Aut(F_{n}) is a subgroup let

Γ′ := Γ/(Γ ∩ Inn(F_{n})

be the corresponding subgroup of Out(F_{n}). Suppose

Hom(F_{n}, G)/G ⟶^{f} W

is a measurable Γ′-invariant mapping. Then f is an ergodic decomposition for Γ′ if and only f ∘ Π is an ergodic decomposition for Γ, where

Hom(F_{n}, G) ⟶^{Π} Hom(F_{n}, G)/G

is the quotient map.

Proof. Since the quotient map Π is absolutely continuous, ergodicity of Out(F_{n}) on Hom(F_{n}, G)/G implies ergodicity of Aut(F_{n}) on Hom(F_{n}, G). Conversely, suppose that Aut(F_{n}) is ergodic on Hom(F_{n}, G). For almost every ρ ∈ Hom(F_{n}, G), the image ρ(F_{n}) is dense in G, and thus Inn(F_{n}) is ergodic on each G-orbit in Hom(F_{n}, G). Thus the quotient map

Hom(F_{n}, G) ⟶ Hom(F_{n}, G)/G

is an ergodic decomposition for the Inn(F_{n})-action. Thus every Aut(F_{n})- invariant function

Hom(F_{n}, G)/G ⟶^{h} ℝ

factors through the quotient map (since it is Inn(F_{n})-invariant) and by assumption is almost everywhere constant. □

In the following lemma, let ι_{j} (for j = 1, …, n) denote the monomorphism defined by

F_{n−1} ↪^{ι_{j}} F_{n}
X_{i} ⟼ \begin{cases}X_{i} if i < j \\ X_{i+1} if i ≥ j \end{cases}

for i = 1, …, n − 1. Denote by

Aut(F_{n−1)} ↪^{I_{j}} Aut(F_{n})

the homomorphism defined by

X_{i} ⟼^{I_{j}(ϕ)} \begin{cases}X_{j} if i = j \\ ι_{j}(ϕ(ι_{j}^{−1}(X_{i}))) if i ≠ j \end{cases}

for ϕ ∈ Aut(F_{n}). Clearly ι_{j} is equivariant with respect to I_{j}.

Lemma 3.2. Let n > 3. Suppose that Aut(F_{n−1}) is ergodic on Hom(F_{n−1}, G). Let 1 ≤ j ≤ n. Then

Hom(F_{n}, G) ⟶^{t_{j}} [−2, 2]
ρ ⟼ tr(ρ(X_{j}))

is an ergodic decomposition for the action of I_{j}(Aut(F_{n−1})) × Inn(G).

Proof. Clearly t_{j} is I_{j}(Aut(F_{n−1})-invariant. Let −2 < τ < 2. The fiber t_{j}^{−1}(τ) identifies with the set of equivalence classes of

((ρ(X_{1}), …, ρ(X_{n−1}), ρ(X_{n}))

where tr(ρ(X_{j})) = τ, that is, the set of Inn(G)-orbits of

Hom(F_{n−1}, G) × tr^{−1}(τ).

By Lemma 3.1 and the ergodicity hypothesis, Aut(F_{n−1}) is ergodic on Hom(F_{n−1}, G). Transitivity of the Inn(G)-action on tr^{−1}(τ) implies ergodicity of Aut(F_{n−1}) × Inn(G) acting on the subset of Hom(F_{n}, G) corresponding to t_{j}^{−1}(τ), which is equivalent to ergodicity of Out(F_{n−1}) on t_{j}−1(τ). □

Suppose inductively that Aut(F_{n−1}) is ergodic on Hom(F_{n−1}, G). By Lemma 3.2, any I_{j}(Aut(F_{n−1}))-invariant measurable function f factors through t_{j} for each j = 1, …, n. We need only consider j = 1 and j = n. Since

Hom(F_{n}, G)/G ⟶^{(t_{1},t_{n})} [−2, 2]^{2}

is onto, any two points ρ, ρ′ in a full measure subset of Hom(F_{n}, G) can be joined by a third ρ′′ such that

t_{1}(ρ′′) = t_{1}(ρ),
t_{n}(ρ′′) = t_{n}(ρ′).

Let f be an Aut(F_{n})-invariant measurable function on Hom(F_{n}, G)/G. Factorization of f through t_{1} implies that

f(ρ) = f(ρ′′)

and factorization of f through t_{n} implies that

f(ρ′′) = f(ρ′).

Thus f(ρ) = f(ρ′′), and f is constant almost everywhere, as desired.

□

4. Other Lie groups

The extension of the proof to products of SU(2) and U(1) proceeds exactly as in [2]. Away from a null subset of Hom(F_{n}, G)/G, the action of the automorphisms of F_{n} preserve tori (products of the U(1) factors and the invariant ellipses). Furthermore on almost every torus, the action is ergodic, and hence every invariant function factors through the trace functions of the generators. The rest of the proof is completely analogous.

References

1. R. Benedetto and W. Goldman, The topology of the relative character variety of the quadruply-punctured sphere, Experimental Mathematics (1999), no. 8:1, 85–104.

2. W. Goldman, Ergodic theory on moduli spaces, Ann. Math. (1997), no. 146, 475–507.

3. , Action of the modular group on real SL(2)-characters of a one-holed torus, Geometry and Topology. (2003), no. 7, 443–486.

4. D. Pickrell and E. Xia, Ergodicity of mapping class group actions on representation varieties, I. Closed surfaces, Comment. Math. Helv. (2001), no. 77, 339–362.

Department of Mathematics, University of Maryland, College Park, MD 20742

  E-mail address: wmg@@math.umd.edu