An ergodic action of the outer automorphism group of a free group

0.33 MB·English

by  William M. Goldman

AN ERGODIC ACTION OF THE OUTER
AUTOMORPHISM GROUP OF A FREE GROUP
6
0
0
WILLIAM M. GOLDMAN
2

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

]
G
D Introduction
.
h Let F be a free group of rank n > 1 and let G be a compact Lie
n
at group. Then Hom(F ,G)admits a naturalvolume formwhich isinvari-
n
m ant under Aut(F ). This volume form descends to a finite measure on
n
[ the character variety Hom(F ,G)/G which is invariant under Out(F ).
n n

3 The purpose of this note is to prove:
v
1 Theorem. Suppose that G is a connected group locally isomorphic to a
0
product of copies of SU(2) and U(1). If n > 2, then the Out(F )-action
4 n
6 on Hom(F ,G)/G is ergodic.
n
0
5 We conjecture that Out(F ) is ergodic on each connected component
n
0 of Hom(F ,G)/G for every compact Lie group G and n > 2.
/ n
h When G = U(1), then this action is just the action of GL(n,Z) on
t
a the n-torus Rn/Zn, which is well known to be ergodic. In fact, certain
m
cyclic subgroups of GL(n,Z) act ergodicly.
:
v The proof relies heavily on [2], both in its outline and a key result.
i
X When n = 2, the action is not ergodic, since it preserves the function
ar Hom(Fn,G)/G −→κ [−2,2]
[ρ] 7−→ tr([X ,X ])
1 2
where X ,X are a pair of free generators for F . However, for each
1 2 2
−2 ≤ t ≤ 2, the action is ergodic on κ−1(t).
Date: February 1, 2008.
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-0405605and DMS-0103889.
1
2 WILLIAMM. GOLDMAN
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 ) is ergodic on
n
Hom(F ,G×G)/(G×G) ←→ Hom(F ,G)/G×Hom(F ,G)/G.
n n n
As in [2], the action of Out(F ) on Hom(F ,G)/G is weak-mixing, that
n n
is:
Corollary. The only invariant finite-dimensional subrepresentation of
the induced unitary representation of Out(F ) on L2(Hom(F ,G)/G)
n n
consists of constants.
I would like to thank David Fisher for pointing out an error in the
original proof of Lemma 3.1 and for many helpful suggestions.
1. Ergodic theory of the SU(2)-character variety
Let {X ,...,X } be a set of free generators for F and let
1 n n
X = X−1...X−1.
0 n 1
Then F is the fundamental group of an n+1-holedsphere S , where
n n+1
the X ,X ,...,X correspond to components of ∂S . The mapping
0 1 n n+1
class group Γ of S embeds in Out(F ) as the subgroup preserving
n+1 n+1 n
the conjugacy classes of the cyclic subgroups hX i for i = 0,...,n.
i
The proof proceeds as follows. Let
(1.1) Hom(F ,G)/G −→f R
n
be an Out(F )-invariant measurable function. We show that f is con-
n
stant almost everywhere.
The main result of [2] applied to the surface S gives the following:
n+1
Proposition 1.1. The mapping
Hom(F ,G)/G −t→∂ [−2,2]n+1
n
tr(ρ(X ))
0
.
[ρ] 7−→ ..
 
tr(ρ(X ))
n
 
is an ergodic decomposition for the action of Γ . That is, for every
n+1
Γ -invariant measurable function
n+1
Hom(F ,G)/G −→h R
n
OUTER AUTOMORPHISMS OF FREE GROUPS 3
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 ֒→ Out(F )
n+1 n
as above, the Out(F )-invariant function f is Γ -invariant, and hence
n n+1
factors through t .
∂
By Proposition 1.1 there exists a function
(1.2) [−2,2]n+1 −→F R.
such that f = F ◦t , where f is the function discussed in (1.1).
∂
2. The case of rank n = 3
First consider the case n = 3. Following the notationof [1, 2], denote
the generators by
A = X , B = X , C = X , D = X
1 2 3 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) ∈ G3
on the generators A,B,(cid:0)C and (cid:1)
ρ(D) = ρ(C)−1ρ(B)−1ρ(A)−1.
2.1. Trace coordinates. The equivalence class [ρ] is determined by
the seven functions
a = tr(ρ(A))
b = tr(ρ(B))
c = tr(ρ(C))
d = tr(ρ(D)) = tr(ρ(D−1)) = tr(ρ(ABC))
x = tr(ρ(AB))
y = tr(ρ(BC))
z = tr(ρ(CA))
4 WILLIAMM. GOLDMAN
subject to the polynomial relation
(2.2) x2+y2 +z2 +xyz =
(ab+cd)x+(ad+bc)y +(ac+bd)z
+(4−a2 −b2 −c2 −d2 −abcd).
In other words, the SL(2,C)-character variety of F is the hypersurface
3
in C7 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 ≥ ∆(a,b,c,d) = 2(a2 +b2 +c2 +d2)−abcd−16 2
−(4−a2)(4−b2)(4−c2)(4−d2)
(cid:0) (cid:1)
of compact components of the cubic surface in R3 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 is the
2
subset V ⊂ R3 defined by traces
3
(x ,x ,x ) ∈ [−2,2]3
1 2 3
satisfying the inequality
(2.3) x2 +x2 +x2 +x x x ≤ 4
1 2 3 1 2 3
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 . Apply
3
(2.2) to x = y,x = a,x = d to see that (a,d,y) ∈ V if and only if y
1 2 3 3
lies in the interval
Y(a,d) := [y (a,d),y (a,d)]
− +
with endpoints
ad± (4−a2)(4−d2)
y (a,d) := .
±
2
p
OUTER AUTOMORPHISMS OF FREE GROUPS 5
For any (a,d) ∈ [−2,2]2, the interval Y(a,d) is nonempty, so that the
restriction of the projection
V −Π−a→,d [−2,2]2
3
a
b
c
a
d 7−→
  d
x (cid:20) (cid:21)
 
y
 
z
 
 
is onto. Furthermore the fiber V (a,d) of the surjection
3
t (V ) Π։a,d [−2,2]2
∂ 3
a
b a
  7−→
c d
(cid:20) (cid:21)
d
 
consists of all (b,c) ∈ [−2,2]2 such that
Y(b,c)∩Y(a,d) 6= ∅.
If −2 < y < 2, then the set of (b,c) such that y ∈ Y(b,c) is the closed
elliptical region E¯ inscribed in the square [−2,2]2 at the four points
y
(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],(cid:8)the fiber Π−1 V3(a,d)(cid:9) equals
(b,c) | y(cid:0)∈ Y(b,c(cid:1)) ,
y∈Y(a,d)
[ (cid:8) (cid:9)
depicted in Figure 4.
6 WILLIAMM. GOLDMAN
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 cor-
respond to the vertices of the tetrahedron.
2
1.5
1
0.5
0
-0.5
-1
-1.5
-1.5 -1 -0.5 0 0.5 1 1.5 2
Figure 2. The SU(2)-character variety has three folia-
tions 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 ver-
tical line segments, respectively.
OUTER AUTOMORPHISMS OF FREE GROUPS 7
2
1.5
1
0.5
0
-0.5
-1
-1.5
-1.5 -1 -0.5 0 0.5 1 1.5 2
Figure 3. For y = −1.2, the set of possible (b,c) ∈
0
[−2,2]2 for which (b,c,y ) is the character of an SU(2)-
0
representation is the interior of the ellipse E inscribed
1.2
in b[−2,2]2 at the four points (±2,∓1.2) and (∓1.2,±2).
2
1.5
1
0.5
0
-0.5
-1
-1.5
-1.5 -1 -0.5 0 0.5 1 1.5 2
Figure 4. Here are the ellipses drawn for the in-
terval Y = [0,1.8]. The union of the closed ellipti-
cal regions is the (b,c)-projection of the V (a,d) where
4
ad = y y = 1.8 and a2 + d2 = (since y = 0), for
+ + −
example a = 1.69,d = 1.062.
8 WILLIAMM. GOLDMAN
2.3. A non-geometric automorphism. Theautomorphismα ∈ Aut(F )
3
defined by:
α
A 7−→ A
B 7−α→ BA−1
α
C 7−→ AC
α
D 7−→ D
induces the following automorphism of the character variety:
a a
b x
c ac−z
α∗
d 7−→ d
   
x ax−b
   
y  y 
   
z  c 
   
Sinceα∗ preservesthecoordinatesa,d,y,thisautomorphismrestricts
to a diffeomorphism on the level sets a = a ,d = d and y = y and
0 0 0
the restriction is linear:
x a −1 0 0 x
0
b α∗ 1 0 0 0 b
  7−→   
c 0 0 a −1 c
0
z 0 0 1 0 z
    
    
The following elementary fact (whose proof is omitted) is useful:
Lemma 2.1. Let −2 < a < 2. The linear transformation
R2 −L→a R2
x ax−y
7−→
y x
(cid:20) (cid:21) (cid:20) (cid:21)
preserves the positive definite quadratic form
x 7−Q→ x2 −axy +y2
y
(cid:20) (cid:21)
and lies in the linear flow generated by the vector field
a a
Υ := x−y ∂ + x− y ∂ .
x y
2 2
(cid:18) (cid:19) (cid:18) (cid:19)
OUTER AUTOMORPHISMS OF FREE GROUPS 9
The trajectories of Υ are the level sets of Q, which are ellipses. L is
a
linearly conjugate to a rotation by angle
(2.4) θ = 2cos−1(a/2).
If θ ∈/ πQ, then L has infinite order and is ergodic on each Q-level
a
set.
The 4-dimensional affine subspaces of R7 corresponding to levels of
(a ,d ,y ) split as products of two affine 2-planes (corresponding to
0 0 0
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 . It preserves the trajectories of the linear vector field
a0
a a
0 0
A = x−b ∂ + x− b ∂
x b
2 2
(cid:18) (cid:19) (cid:18) (cid:19)
a a
0 0
+ c−z ∂ + c− z ∂ .
c z
2 2
(cid:18) (cid:19) (cid:18) (cid:19)
The zeroes of this vector field consist of the origin
x 0
b 0
  =  
c 0
z 0
   
   
and, when a = ±2, the squares defined by
0
x = ±b, z = ±c.
All other trajectories are ellipses. The transformation α∗ acts by rota-
tion along these ellipses through angle θ given by (2.4). When θ/π is
irrational, this action is ergodic. Thus, for almost every a ∈ [−2,2],
0
the restriction of α∗ to these ellipses is ergodic. On a set of full mea-
sure, the function f is constant along the projections of trajectories of
A.
Fix (a ,d ) ∈ (−2,2)2 and consider the equivalence relation ∼ on
0 0
V (a ,d ) generated by the projections of trajectories of A.
3 0 0
Lemma 2.2. For a ,d 6= ±2, all points in V (a ,d ) are ∼-equivalent.
0 0 3 0 0
Proof. Since V (a ,d ) isconnected, itsuffices to prove that eachequiv-
3 0 0
alence class is open. To this end, suppose that (b ,c ) ∈ V (a ,d ); we
0 0 3 0 0
showthatevery(b,c)sufficiently closeto(b ,c )isequivalent to(b ,c ).
0 0 0 0
10 WILLIAMM. GOLDMAN
The imageof the tangent vector A at (x,b,c,z) under the differential
of the coordinate projection
x
b 7−Π→b,c b
c c
(cid:20) (cid:21)
z
 
 
is
a a
0 0
Π A = x− b ∂ + c−z ∂
b,c ∗ 2 b 2 c
(cid:18) (cid:19) (cid:18) (cid:19)
The fiber Π−1(b(cid:0) ,c (cid:1)) is an interval. For some (and hence almost every)
b,c 0 0
(x,z) ∈ Π−1(b ,c ), the vector
b,c 0 0
Π A (b ,x,z,c )
b,c ∗ 0 0
is nonzero. Choose such(cid:0)an (x(cid:1) ,z ). For any open neighborhood U of
0 0
(x ,z ), the values Π A (b ,x,z,c ) for (x,z) ∈ U, span R2.
0 0 b,c ∗ 0 0
Let Φ denote the flow generated by A and choose an open neigh-
t
(cid:0) (cid:1)
borhood U of (b ,x ,z ,c ) in V (a ,d ). The differential of the map
0 0 0 0 0 3 0 0
R× {b }×U ×{c } −→ R2
0 0 0
(t,b ,x,z,c ) 7−→ Π Φ (b ,x,z,c )
(cid:0) 0 0(cid:1) b,c t 0 0
at (0,b ,x ,z ,c ) is onto. The inverse funct(cid:0)ion theorem gu(cid:1)arantees an
0 0 0 0
open neighborhood of (0,b ,x ,z ,c ) mapping onto an open neighbor-
0 0 0 0
hood of (b ,c ), as desired. (cid:3)
0 0
Thus, for almost every (a ,d ) ∈ [−2,2]2, the function F of (1.2) is
0 0
constant along thelevel surfaces Π−1(a ,d ), andhence factorsthrough
a,d 0 0
the projection Π :
a,d
F(a,b,c,d) = F(a,d).
Applying the same argument to the automorphism
γ
A 7−→ CA
γ
B 7−→ B
γ
C 7−→ C
D 7−γ→ DC−1
implies that F factors through the projection Π and F is almost
(b,c)
everywhere constant.