Arboreal Galois Representations
Nigel Boston (boston@math.wisc.edu) and Rafe Jones(jones@math.wisc.edu)
University of Wisconsin-Madison
Abstract.
Let GQ be the absolute Galois group of Q, and let T be the complete rooted d-ary
tree, where d ≥ 2. In this article we study “arboreal” representations of GQ into theautomorphism group of T , particularly in the case d = 2. In doing so, we propose aparallel to the well-developed and powerful theory of linear p-adic representations of
GQ . We first give some methods of constructing arboreal representations and discussa few results of other authors concerning their size in certain special cases. We thendiscuss the analogy between arboreal and linear representations of GQ . Finally, we
present some new examples and conjectures, particularly relating to the question ofwhich subgroups of Aut(T ) can occur as the image of an arboreal representation of
GQ .
Keywords: Galois representation, rooted tree, tree automorphisms, pro-p group,
iterates, monodromy group
MR classification: 11F80, 11R32, 20E08, 20E18

1. Arboreal Representations
To define an arboreal representation, we need two preliminary notions.
For an integer d ≥ 2, the complete rooted d-ary tree is the rooted treewith dn vertices at level n (i.e. distance n from the root), each connected
to d vertices at level n + 1. An automorphism of a tree T with vertex
set V , is a bijection σ : V → V such that v ∈ V is connected to v 0 ∈ V
if and only if σ(v) is connected to σ(v 0 ).
DEFINITION 1.1. An arboreal representation of a profinite group G
is a continuous homomorphism G → Aut(T ), where T is the complete
rooted d-ary tree for some d.
Denote by Tn the set of vertices of the complete rooted d-ary tree
T of distance at most n from the root. This is also the truncation of T
to the first n levels. It is well-known (Nekrashevych, 2005, Proposition
1.4.2) that Aut(Tn ) is isomorphic to the n-fold iterated wreath productof the symmetric group Sd , which we denote by Wn (the value of d will
always be clear from context; usually it will be 2). Moreover, we havethat Aut(T ) ∼= lim Aut(Tn ) ∼= lim Wn , via the natural restriction maps←←
Aut(Tn ) → Aut(Tn−1 ).
c 2006 Kluwer Academic Publishers. Printed in the Netherlands.

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In this article our main interest is in arboreal representations comingfrom iterated polynomials defined over Z. Let f ∈ Z[x] be a polynomialof degree d, and let f 1 = f and f n = f ◦ f n−1 , so that f n is the nthiterate of f . Suppose that all the iterates f n are separable. The successive f -preimages of 0 (contained in Q) now form a complete rooted
d-ary tree. Indeed, for each root α of f n there is a unique root β of f n−1
related to α in the sense that fS(α) = β. Assigning edges according tonthis relation, the disjoint union ∞n=0 {roots of f } becomes a complete
d-ary tree with root 0.
Let Kn (f ) be the splitting field of fn over Q, and let Gn (f ) be thecorresponding Galois group. Consider G(f ) := lim Gn (f ), which is the←
Galois group over Q of the union of the splitting fields of all iterates
of f . Elements of G(f ) are determined completely by their action on
the roots of the f n , and it follows that G(f ) acts faithfully on the tree
formed by these roots. Thus G(f ) gives an arboreal representation ofGQ .
One can generalize this construction to the setting of iterated polynomial covers of P1K , as first done in (Aitken et al., 2005). Let K bea field and t a parameter for P1K . As above, the f -preimages of t form
a complete d-ary rooted tree, and the Galois group corresponding to
the infinite tower of iterations of f acts on this tree. In terms of function fields, the polynomials Fn (x, t) = f n (x) − t ∈ K(t)[x] generate
extensions Kn (f, t) over K(t), and the Galois group G(f, t) of thecompositum of these extensions has a faithful action on the tree ofroots of the Fn (equivalently the tree of f -preimages of t). Moreover,
for any specialization t0 ∈ K of t, the corresponding Galois group
G(f, t0 ) gives an arboreal representation of the absolute Galois group
GK . For details, see (Aitken et al., 2005, p.856). The case discussed in
the previous paragraph corresponds to t0 = 0.
An important attribute of a polynomial is the orbits of its criticalpoints.
DEFINITION 1.2. Let K be a field, let f ∈ K[x] have degree d, andlet γ1 , . . . , γd−1 be the critical points of f . The post-critical set of f is
{f i (γj ) : i ≥ 1, 1 ≤ j ≤ d − 1}. If this set is finite, we call f criticallyfinite, while otherwise we call f critically infinite.
Only primes of K(t) that divide an element of the post-critical set off can ramify in the compositum L of the fields Kn (f, t). A similarstatement holds for specializations of t, although one must add primes
dividing the degree of f . Therefore L/K(t) is finitely ramified if f iscritically finite, and the same statement holds for specializations of t
(Aitken et al., 2005, Theorem 1.1). It follows that G(f, t) is finitely
generated. In section 5 we discuss several instances where f ∈ Z[x] hasarborealrepnsfinal.tex; 7/06/2006; 16:24; p.2

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degree 2 and is critically finite. The resulting finitely generated arboreal
representations furnish an interesting class of examples.

2. Previous Work

Arboreal representations obtained by the method given in section 1
have been studied by a few authors. Let T denote the tree of f preimages of t or some specialization t0 , depending on the context.
When f is critically infinite, most of the investigations have considered
the case t0 = 0 and have shown that the associated arboreal representations are large. Odoni (Odoni, 1985, Theorem 1) considered thefollowing situation: let k have characteristic 0, f be the generic monicpolynomial of degree d defined over K = k(x1 , . . . , xd ), and t0 = 0. He
showed that G(f ) ∼= Aut(T ). Odoni also considered the case K = Q,
t0 = 0, and f = x2 + a, where a ∈ Z, and further work by Stoll (Stoll,1992) showed that G(f ) ∼= Aut(T ) for a > 0, a ≡ 1, 2 (mod 4) andfor a < 0, a ≡ 0 (mod 4). For K = Q, t0 = 0 and f of the form
x2 − ax + a, the second author (Jones, 2006, p. 21) has shown thatG(f ) has finite index in Aut(T ) for all a ∈/ {−2, 2, 4}. It is reasonable
to speculate that whenever a quadratic f ∈ Z[x] is critically infinite,G(f ) will not be finitely generated, and indeed will have finite index
in Aut(T ). Presently neither of these assertions is known (c.f. (Jones,
2006, Conjecture 4.7)).
Another line of inquiry has looked at the case where there is no
specialization and has tended to focus on critically finite f . When
K = C the group G(f, t) is the iterated monodromy group of f (see(Nekrashevych, 2005, Chapter 5)). In the case where f is criticallyfinite, the iterated monodromy group is a finitely generated, self-similar
subgroup of Aut(T ) that is the closure of a group generated by a finite
automaton.
Another interesting example along these lines is due to R. Pink
(personal communication). Let K be a finite field of odd order, andf a quadratic polynomial. Since the critical orbit of f belongs to a
finite extension of K, f must be critically finite, and it follows that
G(f, t) is a finitely generated subgroup of Aut(T ). Little appears to be
known about the subgroups obtained in this manner. For K = F5 andf = x2 − 1, Pink explicitly computed the geometric part ∆ of G(f, t)(meaning we take F5 to be our ground field) and found it has Hausdorff
dimension 2/3. In this case, G(f, t)/∆ is an infinite, procyclic group.

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3. Analogy with p-adic Galois Representations

If O is the valuation ring of some finite extension of Qp and K is
any field, then there are many natural sources of p-adic Galois representations ρ : GK → GLn (O), most notably from algebraic geometry,
namely the Galois action on subquotients of étale cohomology groups ofvarieties defined over K. This is, for instance, how the representations
associated to cuspidal eigenforms are produced. This theory has beenvery fruitful since its inception by Shimura (Shimura, 1966) and others
about 40 years ago, and in particular there are three topics that havedominated the development of the subject.
First, the size of the image has been of interest ever since Serre(Serre, 1972) proved that elliptic curves over Q with no complex multiplication lead to representations with image of finite index (typically1) in GL2 (Zp ). Equivalently, their Lie algebra is gl2 . For more generalabelian varieties, this Lie algebra is conjecturally characterized by the
Mumford-Tate group of the variety.
Second, the images of Frobenius elements under ρ, or at least theircharacteristic polynomials, can be described in geometric terms, yielding some sort of nonabelian reciprocity law. For instance, for a cuspidal
eigenform f of weight k and Nebentypus , the characteristic polynomial of the image of a Frobenius element at q under the p-adic
representation is T 2 − aq T + (q)q k−1 , where aq is the qth coefficientof f . Taking the product of the reciprocals of these characteristic polynomials for varying q with T = q −s and a modified version for primes
q dividing the level of f produces the L-series L(s, ρ), which turns out
to be independent of both ρ and p and so is denoted L(s, f ).
Finally and most recently, the problem of characterizing representations arising from standard constructions has come to the fore. For
instance, elliptic curves and cuspidal eigenforms produce odd representations GQ → GL2 (Fp ), where odd means that the determinantof complex conjugation is −1. Serre’s conjecture (Serre, 1987) proposes that all such odd representations, if absolutely irreducible, shouldarise from some cuspidal eigenform. Most cases of this conjecture haverecently been established. Moreover, the Fontaine-Mazur conjecture(Fontaine and Mazur, 1995), which suggests that the representations
from algebraic geometry should be exactly those that are finitely ramified and potentially semistable at p, has been successively chipped away,most notably by Wiles (Wiles, 1995) (completed by Breuil-ConradDiamond-Taylor (Breuil et al., 2001)), who established enough of it toprove the famous Taniyama-Shimura conjecture that all elliptic curves
over Q are modular.

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In order to carry these three topics over to arboreal Galois representations, we introduce the notion of settledness
DEFINITION 3.1. Given a quadratic polynomial f ∈ Fq [x], a polynomial h ∈ Fq [x] is called f -stable if for every n ≥ 0 h ◦ f n is irreducible.
For a given n let g1 , ..., gr denote the f -stable factors of f n and sn thesum of their degrees. The polynomial f ∈ Fq [x] is called settled if thelimit of sn /2n as n → ∞ is 1.
In (Boston and Jones, 2006) it is shown that if h(a) is not a square
for every a in the critical orbit of f , then h is f -stable. We conjecture
there that every irreducible quadratic f ∈ Fq [x] is settled and give
computational evidence in support.
The notion of settledness is related to arboreal Galois representations because of a result of van der Waerden, stating that if g ∈ Z[x],then (for all but finitely many primes q) the degrees appearing in its
factorization mod q coincide with the cycle structure of the Frobenius
element at q in the Galois group of g over Q.
DEFINITION 3.2. Suppose an element σ ∈ Aut(T ) has image σn ∈Aut(Tn ). A cycle of σn of length 2k is called stable if it is mapped to bya cycle of σr of length 2k+r−n for all r > n. Let the sum of the lengthsof the stable cycles of σn be sn . Then σ is called settled if the limit ofsn /2n as n → ∞ is 1.
For example, an element of Aut(T ) that acts as a single cycle on
every level is settled. Such an element is called an adding machine. Settled elements consist of a proliferation of adding machines on subtreesof T .
The main conjecture of (Boston and Jones, 2006) above implies that
if f is an irreducible quadratic polynomial in Z[x], then the Frobenius
elements in G(f ) are settled. Since by Tchebotarev’s density theorem
the Frobenius elements are dense, it follows that the settled elementsin G(f ) are dense in G(f ). Such a subgroup of Aut(T ) will be called
densely settled. The above notions now allow us to address the threeimportant issues regarding p-adic Galois representations but in the caseof arboreal Galois representations.
First, it is easy to see that settled elements are rare (of densityzero) in Aut(T ). There certainly exist subgroups of Aut(T ) that failto be densely settled, for example because they have too much torsion(torsion elements are never settled). A group is densely settled if andonly if it has a densely settled subgroup of finite index, and so we needconsider densely settled groups only up to commensurability. The next

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section considers some examples. In general, we ask for a classification of them up to commensurability (in analogy to how p-adic Liegroups are classified up to commensurability by their Lie algebra). For
example, which groups defined by automata are densely settled?
Second, assuming the settledness conjecture, we can associate toa Frobenius element at q a certain (possibly infinite) partition of 1.
Namely, if σ ∈ Aut(T ) is its image and the stable parts of σn havedegrees d1 , ..., dr , then d1 /2n + d2 /2n + ... + dr /2n is the initial segmentof what as n → ∞ becomes a partition of 1. This might be thought ofas analogous to a local zeta function. The question arises as to whether
this partition has a finitely expressible generating function. In (Boston
and Jones, 2006), it is conjectured that a Markov process approximates
the factorization of the iterates f n (mod q) and the rate at which thisprocess converges might alternatively be associated to q. The question
then arises as to how to convert these numbers into useful analogues of
L-series.
Finally, the question of which arboreal Galois representations arise
from our construction is one for the future. Some consequences of the
Fontaine-Mazur conjecture are given in the next section.

4. Another Source of Arboreal Representations
The unramified Fontaine-Mazur conjecture (Fontaine and Mazur, 1995)
says that for a number field K any finitely ramified representation ρ :GK → GLn (Zp ) that is unramified at p should have finite image. Theredo, however, exist infinite, finitely and tamely ramified p-extensions of
number fields. The above conjecture implies that their Galois groups
G can have no infinite p-adic analytic quotients.
In fact little is known about these Galois groups. In particular, notone such G has been explicitly presented. The recent breakthroughof Labute (Labute, 2006) shows that some are mild pro-p groups andhence have cohomological dimension 2. In (Boston, 2006) an analogyof the virtual positive Betti number conjecture for 3-manifolds is given,stating that all such G should be virtually Golod-Shafarevich, meaningthat G has a subgroup H of finite index such that r(H) ≤ d(H)2 /4(and H ∼6 Zp ). This conjecture implies the unramified Fontaine-Mazur=
conjecture.
Every finitely generated infinite pro-p group has just-infinite quotients and the above conjecture has strong implications for those of G.
In particular, the Grigorchuk-Wilson dichotomy says that just-infinite
pro-p groups are either branch or have an open subgroup that is a direct
product of finitely many copies of the same hereditarily just-infinite

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pro-p group. Since Golod-Shafarevich groups are never just-infinite,
the conjecture of (Boston, 2006) implies that the just-infinite quotientsof G cannot be of the second kind. They must therefore be branch.
These branch just-infinite quotients of G naturally embed in the automorphism group of a locally finite, rooted tree, so producing arboreal
representations of G.
So far no critically finite polynomial whose iterates produce a tamely
ramified representation is known and so the class of arboreal representations of this section is as yet disjoint from the class considered in therest of the paper. Further discussion of this can be found in (Aitken
et al., 2005).

5. Examples
I. The examples of arboreal representations coming from iterated polynomials defined over Z that were considered in section 2 have densely
settled images. Indeed, their images are of finite index in Aut(T ) and it
follows that they are densely settled. However, the question of whether
Frobenius elements are settled is harder. Consider, for example, f =x2 +1, whose critical orbit over Z is infinite. Modulo 3 all its iterates are
irreducible, so f is f -stable, meaning that the corresponding partition
is just 1/1.
Considering f modulo 7, extensive computations in (Boston andJones, 2006) give evidence for the settledness of Frobenius elements
and indeed the conjecture of an underlying Markov process (here withconvergence rate 0.901). Here we just give the associated partition,
which turns out to be 1/4+1/8+3/16+2/32+8/64+10/128+8/256+
12/512 + 22/1024 + 45/2048 + 45/4096 + 85/8192 + 179/16384 + ...
II. At the other extreme is the example f = x2 − 2. Here the criticalorbit is {−2, 2} and the splitting field of f n is Q(ζ + ζ −1 ), where ζ
is a primitive 2n th root of 1, which has Galois group cyclic of order2n acting regularly on the 2n vertices of T at level n. In the limit, as
n → ∞, we obtain Z2 , the “adding machine”, which is easily seen to
be densely settled. In fact, all nontrivial elements are settled, implyingthat all Frobenius elements are also settled.
III. A somewhat more complicated example is afforded by f = (x−p)2 +p for any odd prime p. Here the critical orbit is just {p} and the Galois
group G(f ) is isomorphic to the group of affine linear transformations of
Z2 , namely A(Z2 ) = {ax + b, a ∈ Z∗2 , b ∈ Z2 }. This group is metabelian
and also the normalizer in Aut(T ) of the group in example II.

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The group A(Z2 ) is densely settled and also all Frobenius elements
are settled. As this group is richer than the adding machine in example
II, we give some details on cycle structures of general elements andFrobenius elements. Given g = ax + b ∈ A(Z2 ), denote by gn the actionof g on Aut(Tn ). There are two fundamental kinds of behavior:
1. If v2 (a − 1) > v2 (b), then for each n all cycles of gn are of equal
length. If g is not the identity and a 6= −1, then for all sufficiently
large n the order of gn+1 is twice that of gn . It follows that g issettled.
2. If v2 (a − 1) ≤ v2 (b), then gn has at least two fixed points for all n.
If g is not the identity and a 6= −1, then for all sufficiently large n
there are nonzero constants c1 , c2 , c3 , and m such that gn consistsof c1 fixed points, c2 2-cycles, c3 4-cycles, c3 8-cycles, . . . , c3 2n−m cycles. Moreover, gn+1 has the same cycle structure, but with anadditional c3 2n−m+1 -cycles. It follows that g is settled.
One sees from this that nearly all elements of A(Z2 ) are settled.
Indeed, the only unsettled elements are the torsion elements, namely
the identity and those g of the form −x + b. It follows immediately thatthe affine group is densely settled.
Now let q be an odd prime not equal to p, and consider a Frobenius
element Frobq at q. Note that for g = ax + b ∈ A(Z2 ), the image of gin the abelianization of A(Z2 ) is simply a. Each h ∈ G(f ) acts on theprimitive 2n th roots of unity by raising to the rn power, and the imageof h in the abelianization of G(f ) is given by lim rn . Since Frobq raises←
elements to the q power, we have that the cycle structure of Frobq isthe same as that of g = qx + b ∈ A(Z2 ) for some b ∈ Z2 . It follows
immediately that Frobq is not torsion, and thus is settled.
We can say even more. Put s = v2 (q − 1). If f s+1 has no roots in
Fq then the cycle structure of Frobq is described by case (1) above,and thus the associated partition of unity is finite. If f s+1 has a root
in Fq then the cycle structure of Frobq is described by case (2), andthe corresponding partition of unity is infinite. Let us do some explicit
calculations in the case p = 3. First suppose that√ q ≡ 3 (mod 4),
which implies that s = 1. If q ≡ 7 √(mod 12) then −3 ∈ Fq and since√−1 ∈ Fq it follows that one of ± −3 is a square in Fq . Hence f 2 =(x − 3)4 + 3 has a root in Fq so that Frobq has an infinite partition. For
example if q = 19 the associated partition is 2/4+2/8+2/16+2/32+....
If q ≡ 11 (mod 12) then f already has no roots in Fq , so f 2 cannot
either, implying the associated partition is finite. For example if q = 23
then the partition associated to q is 4/4.

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Now√ suppose that q ≡ 1 (mod 4), so that s > 1. If q ≡ 5 (mod 12)then −3 6∈ Fq and thus already f has no roots in Fq . It follows that
we are in case (1) and the partition is finite. Indeed, in this case alliterates of f mod q must be irreducible, so the associated partition is
1/1. If q ≡ 1 (mod 12) then f s+1 may either have roots or not haveroots. For instance, if q = 13 then f 3 has no roots, and the associatedpartition is 2/2. For q = 61, f 3 has a root in Fq , and the partition
associated to q is 2/4 + 2/8 + 2/16 + 2/32 + ...
The partitions produced are finite or are geometric series and sohave finite descriptors. One difference with the theory of p-adic representations is the fact that it can happen that the Frobenius elements
for different primes can map to the same conjugacy class in Aut(T ).
For instance, this is the case for the examples of q = 19 and q = 61
above.
IV. An interesting intermediate example is given by f = (x + 1)2 − 2,which has critical orbit {−2, −1}. We calculate the absolute value of thediscriminant of f n to be ∆n = ∆2n−1 2kn where kn = 2n if n is even and
2n + 1 if n is odd. Since ∆n is a power of 2, the splitting field extension
Kn /Q is unramified outside 2 and ∞. Thus Gn := Gal (Kn /Q) isa quotient of the Galois group M of the maximal 2-extension of Q
unramified outside 2 and ∞. The structure of M has been known since1963, when it was proved (Markšaı̆tis, 1963) that M is the free productof C2 and Z2 , i.e. has pro-2 presentation < σ, τ |σ 2 >, where σ canbe taken to be complex conjugation. We would like to determine if thesurjection M → G(f ) is an isomorphism. If so, the fact that the roots of
the iterates of f generate the maximal pro-2 extension of Q unramified
outside {2, ∞} could be viewed as a kind of Jugendtraum.
By Burnside’s basis theorem, M is generated by any two elements
whose images generate M/Φ(M ), which is the maximal elementary∼
abelian quotient of M√. We have
√ M/Φ(M ) = C2 ×C2 , which corresponds
to the extension Q( −1, 2). In particular, any Frobenius element
Frobq for q ≡ 3 or 5 (mod 8) together with σ generates G(f ).
The image of complex conjugation σ in Gn can be taken to be(3, 4)(5, 7)(6, 8)(9, 13)(10, 14)(11, 15)(12, 16)(17, 25)(18, 26)...
i.e. if 2k < i ≤ 2k+1 , then |i − σ(i)| = 2k−1 .
MAGMA allows us to calculate Gn immediately for n = 1, 2, 3, 4.
For n = 1 and n = 2, it is the whole of Wn . For n = 3, Wn has 3
generators and so cannot equal Gn . In fact, the Galois group G3 has
index 2 (so order 64). The 4th iterate f 4 has Galois group G4 of order211 .

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Using van der Waerden’s criterion, we calculate the sequence of cycle
structures of Frobq in successive Wn . For example, for q = 3, we obtain [2], [4], [4, 4], [8, 4, 4], [16, 8, 4, 4], [32, 8, 8, 4, 4, 4, 4], ... The strategy
to determine Gn is to obtain a list A(n) of permissible cycle structure sequences. For n = 5, 6 this is enough to compute Gn exactly. In
particular, it turns out that G5 has order 222 and G6 apparently hasorder 243 . This is confirmed by calculations of Jürgen Klüners, using
his new system that computes Galois groups of any degree.
A closer analysis shows that G5 is generated by the image of σ given
above, together with the element(1, 19, 8, 22, 3, 17, 5, 24, 2, 20, 7, 21, 4, 18, 6, 23)(9, 32, 11, 30, 10, 31, 12, 29)(13, 28, 14, 27)(15, 26)(16, 25)
For n = 6, it is seen that G6 is generated by the image of σ given
above, together with the element(1, 37, 2, 38)(3, 39)(4, 40)(5, 34, 8, 35, 6, 33, 7, 36)(9, 45, 15, 44, 12, 48, 14, 42, 10, 46, 16, 43, 11, 47, 13, 41)(17, 61, 25, 51, 23, 57, 31, 55, 19, 64, 27, 49, 22, 59, 29, 53, 18, 62,
26, 52, 24, 58, 32, 56, 20, 63, 28, 50, 21, 60, 30, 54)
There are some striking patterns regarding the sequence of Gn foundso far – for instance, the nilpotency class of Gn appears to be exactlyn+1
2n−1 . We conjecture that the group Gn has order 2(2 +1)/3 if n is evenn+1
and 2(2 +2)/3 if n is odd. If so, then G(f ) has Hausdorff dimension
2/3. It should be possible then to guess their inverse limit in Aut(T )and see whether this is isomorphic to M .
This compares interestingly with the famous Basilica group, which
also has Hausdorff dimension 2/3. The Basilica group is the iterated
monodromy group (see section 1) of x2 − 1, which is a conjugate of(x + 1)2 − 2.
Acknowledgements
The authors would like to thank Richard Pink and Jürgen Klüners forhelpful discussions.

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