Rank one phenomena for mapping class groups

Benson Farb^{∗}, Alexander Lubotzky^{†}and Yair Minsky^{‡}

March 6, 2000

1 Introduction

Let Σ_{g} be a closed, orientable, connected surface of genus g ≥ 1. The mapping class group Mod(Σ_{g}) is the group Homeo^{+}(Σ_{g})/Homeo_{0}(Σ_{g}) of isotopy classes of orientation-preserving homeomorphisms of Σ_{g}. It has been a recurring theme to compare the group Mod(Σ_{g}) and its action on the Teichmüller space T (Σ_{g}) to lattices in simple Lie groups and their actions on the associated symmetric spaces.

Indeed, the groups Mod(Σ_{g}) share many of the properties of (arithmetic) lattices in semisimple Lie groups. For example they satisfy the Tits alternative, they have finite virtual cohomological dimension, they are residually finite, and each of their solvable subgroups is polycyclic.

A well-known dichotomy among the lattices in simple Lie groups is between lattices in rank one groups and higher-rank lattices, i.e. those lattices in simple Lie groups of R-rank at least two. It is somewhat mysterious whether Mod(Σ_{g}) is similar to the former or the latter. Some higher rank behavior of Mod(Σ_{g}) is indicated by the cusp structure of moduli space, by the fact that Mod(Σ_{g}) has Serre’s property (FA) [CV], and by Ivanov’s version (see, e.g. [Iv2]) for Mod(Σ_{g}) of Tits’s Theorem on automorphism groups of higher rank buildings.

In this note we add two more properties to the list (see, e.g. [Iv1, Iv2, Iv3] and the references therein) of properties which exhibits similarities of Mod(Σ_{g}) with lattices in rank one groups: every infinite order element of Mod(Σ_{g}) has linear growth in the word metric, and Mod(Σ_{g}) is not boundedly generated. In addition, we present a restriction on low-dimensional representations of Mod(Σ_{g}) which has a higher rank flavor.

∗Supported in part by NSF grant DMS 9704640 and by a Sloan Foundation fellowship.

†Supported in part by the US-Israel BSF grant.

‡Supported in part by NSF grant DMS 9971596

Dehn twists have linear growth. In Section 2 we prove that Dehn twists have linear growth with respect to the word metric on the mapping class group of an oriented surface of finite type (Theorem 1.1). This answers a question of Ivanov in [Iv1]. A proof in the case of punctured surfaces was given by Mosher [Mo] using his techniques for constructing an automatic structure on the mapping class group.

This result implies in particular that the embedding of the mapping class group as an orbit in Teichmüller space is not a Lipschitz equivalence (Theorem 2.1). This should be viewed in comparison with the opposite conclusion for higher-rank lattices, by Lubotzky-Mozes-Raghunathan [LMR1].

Theorem 1.1 Let Σ_{g,m} be an oriented surface of genus g and m punctures. Fix a finite generating set for Mod(Σ_{g,m}) and let || · || denote minimal word length with respect to these generators. Then every Dehn twist t has linear growth in Mod(Σ_{g,m}), i.e. there exists a constant c > 0 so that

||t^{n}|| ≥ c|n|

for all n.

Equivalently, one can say that t has positive translation distance in the sense of Gersten and Short [GS], that is

d(t) = lim inf_{n→∞} \frac{||t^{n}||}{n} > 0.

More generally, by previously known results we can conclude the following.

Theorem 1.2 Every element of infinite order in Mod(Σ_{g,m}) has linear growth.

Theorem 1.2 should be compared with [LMR1], which shows that noncocompact, irreducible lattices in higher rank contain elements with logarithmic growth (so-called U-elements). This does not happen for lattices in rank one groups (see [LMR1, LMR2]).

The bounded generation property. Recall that a group Γ is boundedly generated if there is a finite set of elements {a_{1}, …, a_{n}} of Γ so that every element g ∈ Γ can be written

g = a_{1}^{m_{1}} ⋯ a_{n}^{m_{n}}

for some m_{1}, …, m_{n} ∈ Z. In other words, Γ is the (set-theoretic) product of a finite number of cyclic groups. It was shown by Tawgen [Ta] to hold for most (and conjecturally for all) non-cocompact lattices in higher rank simple Lie groups. On the other hand, word-hyperbolic groups are not boundedly generated (see §3.5), and therefore neither are cocompact lattices in rank one groups. It also seems likely that non-cocompact lattices in rank one groups are not boundedly generated.

Theorem 1.3 For g ≥ 1, the group Mod(Σ_{g}) is not boundedly generated.

In fact we prove a stronger result:

Theorem 1.4 For every g ≥ 1 and every prime p, the group Mod(Σ_{g}) has a finite index subgroup H whose pro-p completion Hp̂ is not a p-adic analytic group.

If a group is boundedly generated then so is every finite-index subgroup. Also, the pro-p completion of a boundedly generated group is p-adic analytic. Thus Theorem 1.4 implies Theorem 1.3.

Recall that a pro-finite group is boundedly generated if it is the settheoretic product of finitely many pro-cyclic groups. It is known ([Lu, PR]) that arithmetic groups have the congruence subgroup property if and only if their profinite completion is boundedly generated. This property is known to hold for all non-cocompact, higher-rank lattices and for most cocompact ones. An analogue of the congruence subgroup property for Mod(Σ_{g}) has been conjectured by Ivanov [Iv1]. On the other hand, Theorem 1.4 has the following consequence.

Corollary 1.5 The pro-finite completion of Mod(Σ_{g}) is not boundedly generated as a profinite group

It should be stressed, however, that Corollary 1.5 does not contradict Ivanov’s conjecture, as the equivalence between the congruence subgroup property and bounded generation of the profinite completion is valid only for arithmetic groups.

Low-dimensional representations of Mod(Σ_{g}). It is a well-known open question whether the mapping class group Mod(Σ_{g}), g > 1 has a faithful linear representation ψ : Mod(Σ_{g}) → GL(n, C) for some n ≥ 2. In Section 4 we show that such representations, even for finite index subgroups of Mod(Σ_{g}), do not exist for n < 2\sqrt{g − 1}.

Theorem 1.6 Let H be any finite index subgroup of Mod(Σ_{g}), g ≥ 2. Then there is no faithful representation ψ : H → GL(n, C) for n < 2\sqrt{g − 1}.

2 Growth of Dehn twists

To show that a power T_{α}^{n} of a Dehn twist cannot be expressed as a product of a smaller than linear number of generators, we need a way to measure the “twistiness” of any element of Mod(Σ) around α, in such a way that T_{α}^{n} twists n times and any generator twists a bounded amount. Note that this cannot be made group-theoretically precise, since Mod(Σ) does not admit a non-trivial homomorphism to Z (see, e.g. [Ha])

In the rest of the section we will assume that Σ is hyperbolic of finite volume. This rules out Σ_{0,n} for n ≤ 3, for which Mod(S) is trivial or finite, and the torus Σ_{1,0}, for which Mod(Σ) = SL(2, Z) and the theorem is known. (At any rate since SL(2, Z) is also Mod(Σ_{1,1}) and Mod(Σ_{0,4}) this case is covered).

2.1 The annulus complex and relative twists

Let Y = S^{1} × [0, 1] be an oriented annulus and let A(Y) denote the set of arcs joining S^{1} × {0} to S^{1} × {1}, up to homotopy with endpoints fixed. If a and b in A(Y) do not share any endpoints then we can define their algebraic intersection number a · b. (The arcs inherit an orientation from any fixed orientation on [0, 1], and if this orientation is reversed then the senses of the intersections are unchanged.) We also define a · a = 0.

A lift of any a ∈ A(Y) to Ỹ = R×[0, 1] has endpoints (a_{0}, 0) and (a_{1}, 1), well-defined up to translation (of both) by Z. We can immediately check that, with appropriate choice of orientation of Y,

a · b = ⌊b_{1} − a_{1}⌋ − ⌊b_{0} − a_{0}⌋

(where ⌊x⌋ denotes the largest integer less than or equal to x). It then follows that

a · c = a · b + b · c + Δ  (1)

where Δ ∈ {0, 1, −1}, for any a, b, c ∈ A(Y) such that the intersection numbers are defined.

Now to any essential simple closed curve α in Σ we can associate a closed annulus Y = Y_{α} as follows: Let g_{α} be an isometry of H^{2} representing the conjugacy class of α, and let Y = (\overline{H}^{2} \ F ix(g_{α}))/⟨g_{α}⟩, where \overline{H}^{2} is the standard closed-disk compactification of H^{2}. For any simple curve β in Σ we can consider its lift to int(Y). If β is not homotopic to α then its endpoints are distinct from those of α and hence each component of its lift extends to a properly embedded arc in Y.

Let lift_{α}(β) denote the subset of components that connect the two boundaries of Y, viewed as a subset of A(Y). This set is always finite, and nonempty if β crosses α at least once.

The set of all arcs that arise in this way have mutually disjoint endpoints since any two elements of π_{1}(S) acting on H^{2} with asymptotic axes must be the same up to finite powers. Thus the intersection numbers are well-defined among all these lifts. Let us now define, for α, β and γ simple closed curves in Σ, a subset of Z:

τ_{α}(β, γ) ≡ {b · c : b ∈ lift_{α}(β), c ∈ lift_{α}(γ)}.

(If β or γ is disjoint from α then this is the empty set). Since the components of lift_{α}(β) are mutually disjoint (and similarly for lift_{α}(γ)), we obtain from (1) that diam(τ_{α}(β, γ)) ≤ 2.

Figure 1: The effect of a Dehn twist on the lift to the annulus

Let t = T_{α} be a (leftward) Dehn twist on α. The effect of t^{n} for n > 0 on β, lifted to the components of lift_{α}(β), is to twist each component n times around the annulus, and shift the endpoints a little up on S^{1} × {1} and a little down on S^{1} × {0} (if n < 0 then up and down are interchanged). The shift of endpoints comes from the intersections of lift_{α}(β) with the nonclosed components of the lift of α to Y (see Figure 1). It follows directly that

τ_{α}(β, t^{n}(β)) ⊂ {n, n + 1}  (2)

Furthermore, we have the following properties. If β and γ intersect α, then their intersection number bounds their relative twisting:

max |τ_{α}(β, γ)| ≤ i(β, γ) + 1  (3)

(Here i(·, ·) denotes unoriented geometric intersection number, and by |A| for A ⊂ R we mean {|a| : a ∈ A}.)

To see this, note first that if i(β, γ) = 0 then τ_{α}(β, γ) = {0} so we may assume i(β, γ) ≥ 1. Let b and c be components of lift_{α}(β) and lift_{α}(γ), respectively. We may assume that b and c intersect exactly k = |b · c| times in the annulus. If k = 2 there is nothing to prove, so assume k > 2. Let b_{1}, …, b_{k−1} be successive segments of b bounded by points of b ∩ c, and let c_{1}, …, c_{k−1} be the corresponding segments of c, so that A_{i} = b_{i} ∗ c_{i} is a simple closed curve homotopic to the core of Y, and A_{i} and A_{j} are distinct for i < j. If A_{i} and A_{j} map to the same curve in Σ then the region they bound is identified to make a torus covering Σ, contradicting the assumption that Σ is hyperbolic. Thus all the b_{i} have distinct projections to Σ, and in particular there are at least k−1 intersection points of β with γ. This proves inequality (3).

Finally, for β, γ and δ crossing α we have:

max τ_{α}(β, δ) ≤ max τ_{α}(β, γ) + max τ_{α}(γ, δ) + 2  (4)

and similarly

min τ_{α}(β, δ) ≥ min τ_{α}(β, γ) + min τ_{α}(γ, δ) − 2.  (5)

These follow immediately from applying equation (1) to the components of lift_{α}(β), lift_{α}(γ), and lift_{α}(δ).

2.2 Proof of Theorem 1.1

Let t be a Dehn twist on α. Without loss of generality assume that t is a leftward twist, so that (2) holds.

A set M of simple closed curves binds Σ if every nontrivial, nonperipheral curve in Σ intersects some element of M. Let us choose a particular binding set P as follows: extend α to a pants decomposition of Σ (a maximal collection of disjoint non-homotopic essential curves), and then for each pants curve β add a curve β′ which crosses β once or twice, and misses the other pants curves. Let α′ be the curve in P that crosses α.

Now for any set M of simple closed curves in Σ, define

τ(M) = ⋃_{μ∈M} τ_{α}(α′, μ).

In particular, τ(P) = {0}. Note that τ(M) is always nonempty when M binds Σ.

Let X be some fixed finite generating set for Mod(Σ). Assume without loss of generality that n > 0, and suppose we can write t^{n} as a word w = g_{1} ∘ ⋯ ∘ g_{m}, with each g_{i} ∈ X. Let w_{j} denote the subword g_{1} ∘ ⋯ ∘ g_{j}, and let P_{j} = w_{j}(P). We will prove that

max τ(P_{j+1}) ≤ max τ(P_{j}) + K  (6)

for a fixed K.

Define

B = max{i(β, γ) : β ∈ P, γ ∈ g(P), g ∈ X}

which is finite since the maximum is over a finite set. Since w_{j+1}(P) = w_{j}(g_{j+1}(P)) for any j, and intersection number is Mod(Σ)-invariant, we have for each j, for all β ∈ P_{j} and γ ∈ P_{j+1}, that

i(β, γ) ≤ B.

Inequality (3) then implies that, for any β ∈ P_{j} and γ ∈ P_{j+1} which cross α, max |τ_{α}(β, γ)| ≤ B + 1. Then applying (4), we find that

max τ_{α}(α′, γ) ≤ max τ_{α}(α′, β) + B + 3.

This establishes (6). Now, since we know (using (2)) that τ(w(P)) = τ_{α}(α′, t^{n}(α′)) ⊂ {n, n + 1}, it follows that the length m of w is at least n/(B + 3). This concludes the proof.

Note that the constants can be made explicit if the set of generators X is adapted nicely to P – for example if they are Dehn twists on the components of P.

2.3 Corollaries

Theorem 1.2, that every infinite order element of Mod(Σ) has linear growth, is now easy to put together from known results. The case of simultaneous Dehn twists around a collection of disjoint curves is done in the same way as for a single curve, computing the twisting numbers separately for each component. The case of an element h which is pseudo-Anosov or is reducible and has a power which is pseudo-Anosov on some subsurface is already known – the argument uses the exponential growth of lengths of curves under h (see Mosher [Mo]). By Thurston’s classification of elements of Mod(Σ) [FLP], these are all possible cases.

The second corollary of Theorem 1.1 is the following statement for the Teichmüller space T (Σ) of Σ, on which Mod(Σ) acts properly discontinuously. Fixing a basepoint x_{0} ∈ T (Σ), we may embed Mod(Σ) in T (Σ) as an orbit, i.e. g ↦ g(x_{0}). Endowing T (Σ) with the Teichmüller metric d_{T}, we induce a metric on Mod(Σ) by inclusion. We then have Theorem 2.1 The word metric on Mod(Σ) and the metric induced by inclusion as an orbit in T (Σ) are not Lipschitz equivalent.

Here Lipschitz equivalent means distances in the two metrics are within bounded ratio.

To obtain this result one just needs to know that the translation distance of a Dehn twist in Teichmüller space,

inf_{x∈T (Σ)} d_{T} (x, t(x)),

is zero, and furthermore

d_{T} (x, t^{n}(x)) = O(log |n|)

for any x ∈ T (Σ). In fact Marden-Masur show [MM] that each x is contained in a totally geodesic copy of the hyperbolic plane (a Teichmüller disk) invariant by t, on which t acts as a parabolic isometry.

The situation is similar for non-cocompact lattices in rank-one Lie groups. For example a parabolic element in a non-cocompact lattice in SO(n, 1) = Isom(H^{n}) has linear growth in the word metric but logarithmic growth in the left-invariant metric on SO(n, 1).

By contrast, the situation in a higher-rank semisimple linear Lie group G is the opposite. Lubotzky-Mozes-Raghunathan have shown [LMR1] that for an irreducible lattice Γ in G the word metric and the metric induced from G are Lipschitz-equivalent. For example, a unipotent element of SL(3, Z) has logarithmic growth both in SL(3, R) and in the word metric of SL(3, Z).

As a concrete example, the matrix A = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) has linear growth as

an element of SL(2, Z), but as a matrix in SL(3, Z) via the “upper left hand corner” embedding SL(2, Z) → SL(3, Z) it has logarithmic growth.

3 Pro-p groups and mapping class groups

3.1 Some pro-p preliminaries

Let us briefly recall some definitions from the theory of pro-p groups. Further details and proofs may be found in [DDMS].

A profinite group G is a compact, Hausdorff topological group which is (topologically) isomorphic to an inverse limit of finite groups. Let p be a fixed prime. The profinite group G is a pro-p group if it is topologically isomorphic to an inverse limit of finite p-groups.

The pro-p completion Hp̂ of a group H is defined as the inverse limit

Hp̂ = lim H/N

where the limit is taken over all normal subgroups whose index is a power of p. For example, the p-adic numbers Q_{p} are the pro-p completion of Q.

A profinite group G is finitely generated (as a profinite group) if it contains a finite subset S ⊂ G so that the topological closure of the group generated by S equals all of G.

A profinite group is p-adic analytic if it has the structure of a Lie group over Q_{p}. In particular every such group has a well-defined dimension.

A p-adic analytic profinite group always contains a pro-p subgroup of finite index. See [DDMS] for various characterizations of p-adic analyticity of pro-p groups. A typical example of a compact p-adic analytic group is any closed subgroup of SL(n, Z_{p}); in fact this ends up being all examples (see [DDMS]). In particular, closed subgroups of compact, p-adic analytic groups are analytic. It is also known that an analytic pro-p group is finitely generated (as a pro-p group).

3.2 Analyticity and extensions

In many cases, if the pro-p completion Gp̂ of a group G is taken, then the closure of a subgroup N in Gp̂ is very small with respect to Np̂ (the pro-p completion of N). In other words, the pro-p topology of G induces on N a topology which is much weaker than the pro-p topology of N. The following lemma asserts (when N is f.g. normal in a discrete G) that, after passing to a finite index subgroup L, “most of” Np̂ is preserved.

Lemma 3.1 Let

1 → N → G → H → 1

be an exact sequence of discrete groups. Assume that N is finitely generated. Then G has a finite index subgroup L containing N, so that the closure \overline{N} of N in the pro-p completion of L is mapped onto Np̂/Z(Np̂), the pro-p completion of N modulo its center.

Proof. G acts on N by conjugation. This defines a homomorphism ρ : G → Aut(N). We also have the natural homomorphism ψ : Aut(N) → Aut(Np̂). Since N is finitely generated, the group Np̂ is finitely generated as a pro-p group. As such, it is known (see Theorem 5.6 of [DDMS]) that Aut(Np̂) is a virtually pro-p group, that is, Aut(Np̂) has a normal open pro-p subgroup. In fact

K = Ker(Aut(Np̂) → Aut(Np̂/[Np̂, Np̂]N p ; p̂))

is a normal pro-p subgroup of finite index in Aut(Np̂).

Let L = (ψ ∘ ρ)^{−1}(K). Then L has finite index in G. Moreover, L contains N since N acts trivially on Np̂/[Np̂, Np̂]N p ; p̂. So we have a map ψ ∘ρ from L into the pro-p group K. Now if j : L → Lp̂ is the canonical map from L into its pro-p completion, then we have a map π : Lp̂ → K so that

π ∘ j = (ψ ∘ ρ)|_{L}

Let \overline{N} be the closure of j(N) in Lp̂. Then

π(\overline{N}) = \overline{ψ ∘ ρ(N)}

and it is easy to see that \overline{ψ ∘ ρ(N)} is just equal to the group of inner automorphisms of Np̂, that is:

π(\overline{N}) = \overline{ψ ∘ ρ(N)} = Np̂/Z(Np̂)

Thus \overline{N}, which is a quotient of Np̂, is also mapped onto Np̂/Z(Np̂) and the lemma is proven. ⋄

We will need the following elementary lemma about nilpotent groups.

Lemma 3.2 If Γ is a finitely generated, torsion-free nilpotent group of class k, then the dimension of Γp̂ is at least k.

Proof. The lemma follows easily by induction on k. ⋄

Let us recall that a finitely generated group N is called residually torsionfree nilpotent if, for every g ∈ N, there exists a torsion-free nilpotent group N_{0} and a homomorphism ϕ : N → N_{0} with ϕ(g) ≠ 0.

Lemma 3.3 Let N be a finitely generated residually torsion-free nilpotent group which is not nilpotent. Then for every prime p, the groups Np̂ and Np̂/Z(Np̂) are not p-adic analytic groups.

Proof. Assume Np̂/Z(Np̂) is a p-adic analytic group of dimension r. Let M be any finitely generated torsion-free nilpotent group which is a quotient of N. By our assumptions on N, there are infinitely many such groups M of arbitrarily high nilpotency class, for otherwise N itself would be nilpotent. Let M be such a quotient of N of class at least r + 2. Hence M/Z(M) is of class at least r + 1. By Lemma 3.3 applied to M/Z(M), we have that the dimension of (M/Z(M))p̂ is at least r + 1. But this group is a quotient of Np̂/Z(Np̂), which is of dimension r, a contradiction.

Hence Np̂/Z(Np̂) is not p-adic analytic. Since quotient groups of p-adic analytic groups are p-adic analytic ([DDMS], Theorem 10.7), it follows that Np̂ is not p-adic analytic. ⋄

Combining Lemma 3.1 and Lemma 3.3 gives the following.

Corollary 3.4 Let

1 → N → G → H → 1

be an exact sequence of discrete groups. Assume that N is finitely generated, residually torsion-free nilpotent, but not nilpotent. Then G has a finite index subgroup L containing N so that Lp̂ is not p-adic analytic.

Proof. By Lemma 3.1, G has such a subgroup L for which \overline{N}, the closure of N in Lp̂, is mapped onto Np̂/Z(Np̂). By Lemma 3.3, the group Np̂/Z(Np̂) is not p-adic analytic. Thus \overline{N} and Lp̂ are not p-adic analytic since subgroups and quotient groups of analytic pro-p groups are analytic ([DDMS], Theorem 10.7). ⋄

3.3 Proof of Theorem 1.4

We can now prove Theorem 1.4. First, for g = 1, the group Mod(Σ_{g}) is isomorphic to SL(2, Z), which has a finite index nonabelian free subgroup, and so its pro-p completion is not p-adic analytic and thus not boundedly generated.

We now consider the case g > 2. An old theorem of Dehn states that, for all g ≥ 1, the group Mod(Σ_{g}) is isomorphic to Out^{+}(π_{g}), where π_{g} is the fundamental group of Σ_{g}, Out(π_{g}) is the group of outer automorphisms of π_{g}, and Out^{+}(π_{g}) is a subgroup of index two in Out(π_{g})

The natural action of Homeo(Σ_{g}) on H_{1}(Σ_{g}, Z) clearly descends to an action of Mod(Σ_{g}). This action preserves the pairing on H_{1}(Σ_{g}, Z) = Z^{2g} given by intersection number, so leaves the standard symplectic form on H_{1}(Σ_{g}, Z) invariant, giving the well-known exact sequence:

1 → T (g) → Mod(Σ_{g}) → Sp(2g, Z) → 1

The kernel T (g) is called the Torelli group of genus g. The group T (g) has the following properties:

1. It is finitely generated when g ≥ 3 (see [J]).

2. It is residually torsion-free nilpotent for g ≥ 2 (see [BL]).

3. It is not nilpotent. While this seems to be well-known, and is for example implicit in [Hai], here is a short argument which actually proves the stronger result that T (g) contains a nonabelian free group:

Consider essential, separating, simple closed curves α, β in Σ_{g} with i(α, β) = 2. Then α and β fill out a 4-holed sphere, and the Dehn twists T_{α} and T_{β} lie in the subgroup corresponding to the mapping class group of this subsurface (preserving the holes). We claim that they generate a free group.

To see this, note that the torus covers the sphere with 4 branch points, and the Dehn twists on α and β in the sphere lift to squares of Dehn twists on the torus. That is, they correspond to the matrices

( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) and ( \begin{matrix}1 & 0 \\ 2 & 1 \end{matrix} ) in PSL(2, Z)

These generate a free group, as one can easily see by looking at an appropriate fundamental domain in H^{2} and applying the standard Schottky argument.

Applying Corollary 3.4 to the exact sequence above now gives the conclusion of Theorem 1.4 for g ≥ 3.

For g = 2 the above proof breaks down since the Torelli group T (2) is not finitely generated. In fact it was shown by Mess [Me] that T (2) is an infinitely generated free group. We can still modify the arguments above to cover the case g = 2 as well. We sketch here a proof of how to do this.

For each prime p, let π_{p} denote the composition

Mod(Σ_{g}) → Sp(2g, Z) → Sp(2g, Z/pZ)

and let L = ker(π_{p}). Consider Lp̂. Let \overline{N} be the closure of the Torelli group in Lp̂. If \overline{N} is not finitely generated we are done, i.e. Lp̂ cannot be p-adic analytic since every closed subgroup of an analytic group is finitely generated. So suppose that \overline{N} is finitely generated. We will show that \overline{N} has quotients of arbitrarily large dimension, so that \overline{N} (hence Lp̂) is not p-adic analytic.

To this end, consider the image of \overline{N} in Out((Γ_{1}/Γ_{i})p̂), where Γ_{i} is defined inductively as follows: Γ_{1} = π_{1}(Σ_{g}) and Γ_{i+1} = [Γ_{1}, Γ_{i}] for i > 1. Consider the pro-p completion (Γ_{1}/Γ_{i})p̂. As Γ_{i} is characteristic in Γ_{1}, there is a homomorphism

Out(Γ_{1}) → Out(Γ_{1}/Γ_{i}) → Out((Γ_{1}/Γ_{i})p̂)

which induces a homomorphism ψ from L to a pro-p subgroup of Out((Γ_{1}/Γ_{i})p̂). As the image of ψ is contained in a pro-p group, we have that ψ can be extended to Lp̂. One can then check, á la the analysis in Andreadakis [An], that the image of \overline{N} in Out((Γ_{1}/Γ_{i})p̂) is a nilpotent analytic group whose dimension is growing to infinity with i (since N is not nilpotent). This proves that \overline{N} is not finite dimensional, i.e. is not analytic.

⋄

3.4 Automorphism groups of free groups

Let Aut(F_{n}) (resp. Out(F_{n})) denote the automorphism group (resp. outer automorphism group) of the free group F_{n} of rank n. The same technique as was used to prove Theorem 1.4 and Theorem 1.3 applies to Aut(F_{n}) and Out(F_{n}).

Theorem 3.5 For every n ≥ 2 and every prime p, the group Aut(F_{n}) has a finite index subgroup L whose pro-p completion Lp̂ is not p-adic analytic; similarly for Out(F_{n}). In particular neither Aut(F_{n}) nor Out(F_{n}) is boundedly generated.

Remark. The latter result for Aut(F_{n}) was proved by Sury [Su].

Proof. We describe the proof for Out(F_{n}). For n ≥ 3 the proof is exactly the same as the case Mod(Σ_{g}) = Out(π_{1}(Σ_{g})). The corresponding Torelli subgroup

T = ker(Out(F_{n}) → SL(n, Z))

is indeed finitely generated and residually torsion-free nilpotent (see [BL] and the references therein). Further, T is not nilpotent. One can do this explicitly (as in the case for the Torelli subgroup of Mod(Σ_{g}) above), or for n ≥ 4 as follows: if T were nilpotent then one could deduce from [Lu2] (which holds for n ≥ 4) that Out(F_{n}) is linear, which contradicts [FP]. Applying Corollary 3.4 now finishes the proof.

For n = 2, the group Out(F_{n}) is isomorphic to GL(2, Z), which has a nonabelian free subgroup of finite index, hence its pro-p completion is not analytic. In particular it is not boundedly generated. ⋄

3.5 Word-hyperbolic groups are not boundedly generated

We would like to record the following.

Proposition 3.6 A non-elementary word-hyperbolic group is not boundedly generated.

Proof. Gromov has shown that every non-elementary word-hyperbolic group G has an infinite torsion quotient H. If G were boundedly generated then so would H be. But it is clear that a boundedly generated torsion group must be finite. ⋄

4 Low-dimensional representations of Mod(Σ_{g})

In this section we prove Theorem 1.6.

Let C = {C_{1}, …, C_{2g}} be a collection of essential, homotopically distinct, simple closed curves such that i(C_{i}, C_{g+i}) = i(C_{g+i}, C_{i}) = 1 for each 1 ≤ i ≤ g and i(C_{i}, C_{j}) = 0 otherwise.

For any simple closed curve α, let T_{α} denote the Dehn twist about α. Then i(α, β) = 0 if and only if T_{α} commutes with T_{β}. In fact (see, e.g. [Iv2], Theorem 7.5C), powers T_{α}^{m}, T_{β}^{n}, m, n ∈ Z^{+} commute if and only if i(α, β) = 0

Now let any finite index subgroup H of Mod(Σ_{g}) be given. As H has finite index, there exists l > 0 so that Y_{C_{i}}^{l} ∈ H for all 1 ≤ i ≤ 2g.

For any subset S ⊂ C we denote by T_{S}^{l} the subgroup of Mod(Σ_{g}) generated by {T_{C}^{l} : C ∈ S}. Let

S = {S ⊂ C : T_{S}^{l} < H is abelian} = {S ⊂ C : if α, β ∈ S then i(α, β) = 0}

Now let

N = max_{S∈S} dim(\overline{ψ(T_{S}^{l})})

where \overline{H} denotes the Zariski closure of the subgroup H in GL(n, C) and dim denotes the dimension of an algebraic group. Let B = {C_{i_{1}}, …, C_{i_{r}}} be an element of S which both realizes this maximum and such that r is minimal over all elements of S realizing this maximum.

We claim that r ≤ N. As \overline{ψ(T_{B}^{l})} is abelian, it is enough to show that, for an abelian algebraic group A = \overline{< a_{1}, …, a_{r} >} where the set \{a_{i}\} is minimal as above, that r ≤ dim(A). To see this, one proceeds by induction on r, the case r = 1 being clear. By the minimality assumption, A_{2} = \overline{< a_{1}, …, a_{r−1} >} has dimension at most N −1, for otherwise we could throw away a_{r}. As the product of algebraic groups is algebraic (see, e.g. [Sp], p.31), the Zariski closure of a product is the product of Zariski closures. Hence {a_{1}, …, a_{r−1}} is minimal for A_{2}, so by induction r − 1 ≤ N − 1. Hence r ≤ N.

Since in an algebraic group the centralizer of a single element is clearly algebraic, and the intersection of algebraic groups is algebraic, we see that the Zariski closure of a finitely-generated abelian group is also abelian. In particular \overline{ψ(T_{B}^{l})} is an abelian algebraic subgroup of GL(n, C). By an old result of Schur [Sc] any such subgroup of SL(n, C) has dimension at most ⌊n^{2}/4⌋ ≤ n^{2}/4. But any maximal abelian subgroup of GL(n, C) is of the form ZA, where A is such a subgroup on SL(n, C) and Z denotes the center of GL(n, C). Hence any abelian algebraic subgroup of GL(n, C), in particular \overline{ψ(T_{B}^{l})}, has dimension at most n^{2}/4 + 1.

We now clearly have

r ≤ N ≤ n^{2}/4 + 1

so the hypothesis n < 2\sqrt{g − 1} gives

g > n^{2}/4 + 1 ≥ r

Hence there are curves α_{1}, α_{2} ∈ C so that :

(a) i(α_{i}, γ) = 0 for i = 1, 2 and for each γ ∈ B.

(b) i(α_{1}, α_{2}) = 1.

Recall that powers T_{α}^{m}, T_{β}^{n}, m, n ∈ Z^{+} commute if and only if i(α, β) = 0. By (a), each T_{α_{i}} commutes with T_{B}^{l}. By the maximality property of B, for i = 1, 2 the abelian algebraic group \overline{ψ(T_{B}^{l} ∪ T_{α_{i}})} has dimension N, and so must contain \overline{ψ(T_{B}^{l})} as a finite-index subgroup. In particular there exist positive integers m_{1}, m_{2} so that

ψ(T_{α_{i}})^{m_{i}} ∈ \overline{ψ(T_{B}^{l})} for i = 1, 2

Hence ψ(T_{α_{1}}^{m_{1}}) and ψ(T_{α_{2}}^{m_{2}}) commute. But by (b) we have that no power of T_{α_{1}} commutes with any power of T_{α_{2}}. Hence it must be that ψ is not faithful. ⋄

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Benson Farb: Dept. of Mathematics, University of Chicago 5734 University Ave. Chicago, Il 60637 E-mail: farb@math.uchicago.edu

Alex Lubotzky: Department of Mathematics Hebrew University Jerusalem, Israel E-mail: alexlub@math.huji.ac.il

Yair Minsky: Department of Mathematics SUNY Stony Brook Stony Brook, NY 11794 E-mail: yair@math.suysb.edu