THE IMAGE OF AN ARBOREAL GALOISREPRESENTATIONNIGEL BOSTON AND RAFE JONES
Abstract. Much is known regarding images of p-adic Galois representations coming from subquotients of étale cohomology groups of varieties
over number fields. In particular, the Mumford-Tate conjecture gives
them up to subgroups of finite index, and has been proved in manycases by Serre. In the analogous situation of arboreal Galois representations little is known. In this paper we give a conjectural descriptionof their images in the case that they arise via iteration of a given polynomial. We also discuss in depth the case of iteration of a quadratic
polynomial with integer coefficients.

1. Arboreal Representations
There are two main sources of totally disconnected, locally compact groups,
namely matrix groups over local fields and automorphism groups of locallyfinite trees [22]. Continuous homomorphisms of Galois groups into the former have been well studied and exploited but those into the latter havebarely been touched upon.
If K is a number field, S a finite set of primes of K, K S a maximal
algebraic extension of K unramified outside S, and G =Gal(K S /K), theFontaine-Mazur conjecture [6] characterizes those representations G → GLn (Zp )coming from algebraic geometry by the group-theoretical condition of being
potentially semistable at p. In the case that S contains no prime lying abovep, the conjecture states that every such representation should have finite image. This state of affairs is unsatisfactory in that for many K and S thegroup G has an infinite pro-p quotient, which therefore cannot be seen bya p-adic Galois representation. A refinement, the virtual Golod-Shafarevich
conjecture [2], implies that in such cases there should exist representations
G →Aut(T ) with image having nonzero Hausdorff dimension (see below for1

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NIGEL BOSTON AND RAFE JONES

its definition), where T is a locally finite, rooted tree. We call such representations arboreal.
This source of arboreal Galois representations is hard to control. In particular, there is not one infinite finitely, tamely ramified p-extension whose
Galois group has been explicitly presented. A more explicit source of arboreal Galois representations is given by the Galois action on the roots of theiterates of a given polynomial. In this paper we consider the images of such
representations, mostly in the case of quadratic polynomials.
Let f be a degree d polynomial in Q[x], all of whose iterates are separable.
The roots of the nth iterate f n of f can be identified with the dn vertices
at level n of the d-ary rooted tree T , in such a way that the Galois groupof f n embeds in Aut(Tn ) where Tn consists of the subtree of vertices up to
and including level n. Putting these embeddings together yields a continuous
homomorphism GQ := Gal(Q/Q) →Aut(T ), whose image G(f ) is the Galois
group over Q of the extension generated by all roots of all iterates of f . The
Galois group of f n will be denoted Gn (f ). All Gn (f ) will be transitive if
and only if every iterate of f is irreducible, in which case (as is typical inthis paper) we call G(f ) spherically transitive.
The main focus of this paper is the identification of and properties of G(f )as a subgroup of Aut(T ). In particular, we focus on the case d = 2. In thiscase, Aut(Tn ) is a 2-group, in fact a Sylow 2-subgroup of Sym(2n ) so of ordern22 −1 , whence Gn (f ) is a 2-group. The main measure of the size of G(f ) is
its Hausdorff dimension, given by the liminf of log2 (|Gn (f )|/(2n − 1)).
After presenting some analogies with p-adic representations in section 2,
we discuss some general properties that G(f ) must obey in section 3. In
Section 4 we examine the possibilities for G(f ) when f ∈ Z[x] is quadratic.
We decompose the set of such f into natural families and compute G(f )generically in each of these families. We then examine the behavior withineach family, conjecturing that G(f ) must have finite index in the genericgroup. In section 5 we summarize our main conjectures.

THE IMAGE OF AN ARBOREAL GALOIS REPRESENTATION

3

2. Analogies with p-adic Galois Representations
If V is a smooth projective variety over Q, then its étale cohomology
groups are p-adic vector spaces on which GQ acts. These and their subquotients yield representations GQ → GLn (Qp ), which we say come fromalgebraic geometry. Since there is always a stable lattice, this in turn produces representations GQ → GLn (Zp ). The most important examples are
the 1-dimensional (cyclotomic) representations given by Galois action on thepn th roots of 1, the 2-dimensional representations given by Galois action on
the pn -division points of an elliptic curve, and the 2-dimensional representations associated to modular forms arising as subquotients of the Galois
action on the pn -division points of Jacobians of modular curves.
In analyzing these representations the main questions concern the size ofthe image and the nature of the images of Frobenius elements. In general, forrepresentations coming from algebraic geometry, the image is conjecturally
determined up to finite index subgroups by the p-adic points of a Hodge(or Mumford-Tate) group. The first such results concerned CM elliptic
curves, whose Galois representations have image the normalizer of a Cartan
subgroup of GL2 (Zp ). Serre [17] showed that elliptic curves without CM
have Galois representations whose image is of finite index in GL2 (Zp ). Othercases of the Mumford-Tate conjecture have been established by Deligne [5],Serre [18], [19], [20], and Chi [4].
Analogues for Galois representations associated to Drinfeld modules havebeen discovered by Pink [15]. Our goal is to shed light on these same questions in the case of arboreal Galois representations associated to the iterates
of a given polynomial and to observe analogous phenomena.
3. Densely settled subgroups
Let the notation be as on p. 2. We focus on the case where f is a polynomial with integer coefficients. Let ρf denote the continuous homomorphism
Gal (Q/Q) −→ Aut(T ) whose image is G(f ). In this section we discuss
some properties of Im ρf , particularly in the case where f is quadratic, andgive conjectures on other properties.

4

NIGEL BOSTON AND RAFE JONESIn [3] we conjecture that the image of ρf must possess a property known
as settledness that depends on the cycle decomposition of the action of Im ρfon each level of T . Denote by Vn the vertices of T that are distance n fromthe root, or in other words the roots of f n . Let σ ∈ Im ρf , and denote by σnthe restriction of σ to Vn . Call a cycle C of σn stable if for each m ≥ n, theportion of σm that restricts to C is a single cycle of length 2m−n · |C|, where|C| denotes the length of C. We say that σ is settled if the proportion ofelements of Vn contained in stable cycles of σn goes to one as n grows. We
say that Im ρf is densely settled if settled elements are dense in the profinite
topology.
Let Frobp be a Frobenius element at the prime p. Then the cycle decomposition of the action of Frobp on the roots of f n is given by the degreesof the irreducible factorization of f n mod p. Since Frobenius elements aredense in Im ρf by the Tchebotarev Density Theorem, we can show thatIm ρf is densely settled by showing that the irreducible factors of f n mod p
are settled in a sense precisely analogous to the definition given in the previous paragraph. In [3] we conjecture that any irreducible quadratic overFp is settled, and we give a heuristic argument and extensive computations
in support of this conjecture. Indeed it seems likely that all separable polynomials over Fp are settled. From this it would follow that Im ρf is densely
settled for all quadratic f ∈ Z[x].
However, while conjecturally Im ρf must be settled at least for quadratic
f , this alone appears insufficient to characterize the possible images of ρf .
Indeed, there are many abelian subgroups of Aut(T ) that are settled, butvery few can occur as Im ρf for quadratic f ∈ Z[x]:Call f critically infinite if the set {f n (γ) : n = 1, 2, . . .}, where γ is thecritical point of f , is infinite.
Theorem 3.1. Let f ∈ Z[x] be a critically infinite quadratic polynomialwith all iterates irreducible. If G(f ) is abelian, then it is cyclic.
Proof: Suppose that G(f ) is abelian, and denote by Gn (f ) the action ofG(f ) on Vn . We claim that the center of Gn (f ) contains a unique element

THE IMAGE OF AN ARBOREAL GALOIS REPRESENTATION

5of order 2 for all but finitely many n. This claim immediately implies thatGn (f ) is cyclic for infinitely many n, and thus for all n. Hence G(f ) is cyclic.
To prove the claim, first note that in [9, Corollary 4.11 and remarks before]
it is shown that if f n is irreducible and Disc f n is not a square in Q, thenthe center of Gn (f ) contains a unique element of order 2. From [10, p. 10]
we have that the squarefree part of Disc f n is equal to the squarefree part off n (γ). If there are infinitely many n with y 2 = f n (γ) for some y ∈ Q, thensince f is critically infinite, it follows that for all k > 0 there are infinitely
many distinct (x, y) ∈ Q2 with y 2 = f k (x). Since f is assumed separable, kcan be selected so the curve Y 2 = f k (X) has genus at least two. However,this violates Faltings’ theorem [8, Part E].
As for quadratic f such that {f n (γ) : n = 1, 2, . . .} is finite (such f
are called critically finite), one can reproduce the proof of Theorem 3.1
provided that Disc f n is not a square for infinitely many n. A straightforward
calculation shows that all critically finite quadratic f ∈ Z[x] are of the form(x − k)2 + k, (x − k)2 + k − 1, or (x − k)2 + k − 2 for some k ∈ Z. In each ofthese cases, Disc f n is a square for all sufficiently large n only if k is a square,k = 1, or k−2 is a square respectively. For these values of f and k, the group
Gn (f ) appears to be either cyclic, isomorphic to (Z/2n Z)∗ , or nonabelian.
This follows from the first part of section 4.2 as well an examination of thecases f = (x − k)2 + k, k = ±1 and (x − k)2 + k − 2, |k| ≤ 2 and the factthat any finite index subgroup of Z2 o Z∗2 is nonabelian. Thus even when
f is critically finite, only a very restricted set of abelian groups can occuras G(f ). This leads us to ask what additional properties besides settledness
G(f ) should possess.

4. Quadratic Polynomials
4.1. Generic quadratic families. We now turn to a more detailed analysis
of Im ρf in the case of quadratic f ∈ Z[x]. Any such f may be written(1)

f = (x − γ)2 + γ + m,

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NIGEL BOSTON AND RAFE JONES

where 2γ and 4m are integers. One easily checks that f is critically finite
if and only if m ∈ {−2, −1, 0}. We now fix a value of m and consider all fwhose decomposition in (1) has the prescribed value of m. Let us describe
the Galois groups for iterates of a generic such f .
Theorem 4.1. Suppose f = (x − t)2 + t + m ∈ Q(t)[x], where m 6∈{0, −1, −2}. Let Gn (f ) be the Galois group over Q(t) of the splitting field of
the nth iterate f n , and let G(f ) = lim Gn (f ). Then G(f ) ∼= Aut(T ), where←

T is the infinite binary rooted tree of preimages of 0.
Proof: We first show that f n is irreducible, and therefore separable, for alln. From [10, Proposition 3.3] we need only show that f n (t) is never a squarein Q(t). This clearly holds, as f n (t) has degree one for each n.
We now wish to show that for all n, Gn (f ) is isomorphic to Aut(Tn ), thegroup of automorphisms of the binary rooted tree of height n. It is wellknown that Aut(Tn ) is isomorphic to the n-fold iterated wreath product ofn

Z/2Z, and in particular |Aut(Tn )| = 22 −1 . Let Kn denote the splitting
field of f n over Q(t), and put Hn = Gal (Kn /Kn−1 ). If we show thatn−1

|Hn | = 22
for all n, then it follows by induction and comparison of degrees∼
that Gn = Aut(Tn ).
The argument in [10, Lemma 4.1] applies verbatim over Q(t), and puttingg = id in that Lemma shows that |Hn | = 22

n−1

if and only if f n (t) is not asquare in Kn−1 . Now the only primes of Q(t) (by which we mean a discrete
valuation ring OP with maximal ideal P ) that ramify in Kn are those suchthat Disc Kn is zero in OP [16, Proposition 7.9] and moreover if Disc f n is
not zero in OP then Disc Kn is not zero in OP (for the last assertion one canadapt the argument in [12, Corollary 2, p. 157]). It follows that the only
primes ramifying in Kn−1 are those with Disc f n−1 = 0 in OP , and by [10,p. 10] these are precisely the primes dividing f m (t) for some m ≤ n − 1.
To show that Disc f n is not a square in Kn−1 , we show that its squarefreepart is divisible by a prime that does not ramify in Kn−1 . By [10, p. 10]
the squarefree part of Disc f n is precisely the squarefree part of f n (t). In
the case currently under consideration, f n (t) has degree 1 as a polynomial

THE IMAGE OF AN ARBOREAL GALOIS REPRESENTATION

7

in t, and thus is squarefree. Moreover, since f is critically infinite by ourassumption that m 6∈ {0, −1, −2}, the set {f m (t) : m = 1, 2, . . .} consistsof infinitely many distinct degree 1 polynomials, and thus all its elements
are relatively prime to one another. It follows that f n (t) cannot divide
Disc f n−1 , and thus does not ramify in Kn−1 .

We remark that if K is a number field and OK is its ring of integers, thenas in (1) every quadratic polynomial in OK [x] can be written as (x−γ)2 +γ +m, where 2γ, 4m ∈ OK . Moreover, if m is such that f = (x − t)2 + t + m ∈K(t)[x] is critically infinite, a straightforward adaptation of the proof of
Theorem 4.1 shows that G(f ) ∼= Aut(T ).
Theorem 4.2. Using the notation of Theorem 4.1, suppose f = (x − t)2 +t + m ∈ Q(t)[x], where m ∈ {0, −2}. Then G(f ) ∼= Z2 o Z∗ , the group of2

invertible affine linear transformations of Z2 .
n

Proof: In the case m = 0, we have f n = x2 − t, whence the splitting√nfield Kn of f n over Q(t) is just Q( 2 t, ζ2n ). One now checks easily that f nremains irreducible over Q(t, ζ2n ), and thus [Kn : Q(t)] = 22n−1 . Therefore
√
√
n
n b
the map ζ2n 7→ ζ2an , 2 t 7→ 2 t is an automorphism of Kn /Q(t) for alla ∈ Z/2n Z∗ and all b ∈ Z/2n Z. It follows that Gal (Kn /Q(t)) is isomorphic
to the group of invertible affine linear transformations of Z/2n Z and thusG(f ) is isomorphic to the same group over Z2 .
If m = −2, the proof is identical to that of part 1) of Theorem 1.1 in [7].
Any extension L of Q(t) has a sub-extension A = Q ∩ L of Q, which wecall the arithmetic part of L (it is also the maximal constant field extension
contained in L). We refer to the extension L/A(t) as the geometric part of L.
If L is the splitting field of a polynomial g, there is a natural isomorphism of
L/A(t) and E/C(t), where E is the splitting field over C(t) of g considered
as a polynomial in C(t)[x]. It now follows that if Kn is the splitting fieldover Q(t) of f n for some f ∈ Q(t)[x], L = ∪n Kn , and E/C(t) corresponds to
L/A(t), then the group Gal (E/C(t)) is isomorphic to the closure in Aut(T )8

NIGEL BOSTON AND RAFE JONESof the iterated monodromy group of f over C (see [13, Chapter 5]). This
last group may be computed using geometric methods. In the case of f =(x − t)2 + t + m ∈ Q(t)[x], m 6∈ {0, −1, −2}, the geometric part of L has
Galois group Aut(T ), while if m ∈ {0, −2} then the geometric part of L has
Galois group Z2 . In the former case, this observation can be used to provide
an alternate proof of Theorem 4.1
We now arrive at the lone m not treated in Theorems 4.1 and 4.2, namelym = −1. If f = (x − t)2 + t − 1, Kn is the splitting field over Q(t) of f n ,and L = ∪n Kn , then Gal (L/A(t)) is isomorphic to the closure B of theBasilica group in Aut(T ). The Basilica group is a certain group generated
by a finite automaton, and about which much is known [1] and which has
a natural action on T . Unfortunately, much less seems to be known about
B. Thus G(f ) has B as a normal subgroup, with quotient Gal (A/Q). It
remains an open problem to determine Gal (A/Q) and to find the structureof G(f ) beyond the previous observation.
4.2. Quadratic Specializations. In the previous section, we examined
generic behavior of ρf within certain families of quadratic f . In this section
we focus on the behavior of ρf for specific f . The general theme is thatspecializations of t in Theorems 4.1 and 4.2 should yield Galois groups offinite index in the generic groups described in Theorems 4.1 and 4.2. Thisis reminiscent of work of Serre (e.g. [17]), where the Lie algebra of the Tate
module of an elliptic curve determines the Tate module up to finite index.
Lemma 4.3. Let F be a field containing a primitive kth root of unity ζk ,and let α ∈ F . Then [F (α1/k ) : F ] = k/d, where 1 ≤ d ≤ k is maximal suchthat α1/d ∈ F .
0

Proof: Let e := [F (α1/k ) : F ] and note that we have αe /k 6∈ F for all000

e0 < e. Indeed, if αe /k ∈ F then α1/k is a root of xe − αe /k , whence[F (α1/k ) : F ] ≤ e0 , a contradiction.
Now let f (x) ∈ F [x] be an irreducible factor of xk − α, and let β be a rootof f . Write β = ζki α1/k for some 1 ≤ i ≤ k, and note that F (β) = F (α1/k )THE IMAGE OF AN ARBOREAL GALOIS REPRESENTATION

9since F contains ζk . Thus Gal (F (β)/F ) is cyclic of order e. It follows thatf (x) =e Yi+(jk/e) 1/k

x − ζkα= xe − ζkei αe/k ,j=1

where the second equality is obtained by comparing roots. Thus αe/k ∈ F ,proving that 1/d = e/k.



Theorem 4.4. Let t0 ∈ Z, let f = (x − t0 )2 + t0 + m ∈ Q[x], where m = 0.
Let Gn (f ) be the Galois group over Q of the splitting field of the nth iterate
f n , and let G(f ) = lim Gn (f ). Then G(f ) is a finite-index subgroup of←

Z2 o Z∗2 unless t0 ∈ {−1, 0, 1}.
Proof: Note that Z2 o Z∗2 = lim Z/2n Z o (Z/2n Z)∗ , and the order of←

Z/2n Zo(Z/2n Z)∗ is 22n−1 . Thus it suffices to show that [Z/2n Zo(Z/2n Z)∗ :Gn (f )] is bounded as n goes to infinity.
nNote that f n (x) = (x − t0 )2 + t0 , and the splitting field Kn of f n over Q
must contain a primitive 2n th root of unity ζ2n as long as t 6= 0. By Lemma
4.3, the degree [Kn : Q(ζ2n )] is 2n /d, where 1 ≤ d ≤ 2n is maximal such that
1/dt0

∈ Q(ζ2n ). Thus [Kn : Q] = 22n−1 /d, implying that [Z/2n Zo(Z/2n Z)∗ :Gn (f )] = d. Now provided that t0 6= ±1, there is e > 0 and a prime p such
1/ethat vp (t0 ) is odd. If p 6= 2 then p does not ramify in Q(ζ2n ), and it follows√
1/(e+1)that t0
is not in Q(ζ2n ). If p = 2, then one notes that 4 2 6∈ Q(ζ2n )√(otherwise Q( 4 2) would be a subfield of an abelian extension, and thus
1/(e+2)

Galois, which is a contradiction). Therefore t0

6∈ Q(ζ2n ). In eithercase we have d ≤ e + 2. Thus d is independent of n for n sufficiently large,
which proves the Theorem.



We note that if m = −2, then matters are similar to the case m =0, although more complicated and with more exceptional values, namelyt0 ∈ {−2, −1, 0, 1, 2}. In this case, the roots of f n generate a subfield ofndegree at most two of the splitting field over Q of x2 − β, where β is anroot of x2 − t0 x + 1 [7, p.8]. Let α be a root of x2 − β, and note that if[Z/2n Z o (Z/2n Z)∗ : Gal (Q(α, ζ2n ))/Q(β)] is bounded as n goes to infinity,

10

NIGEL BOSTON AND RAFE JONES

then it follows that G(f ) has finite index in Z2 o Z∗2 . By Lemma 4.3 it is√nenough to show that 2 β 6∈ Q(β)(ζ2∞ ) for some n > 0, unless β is a rootof unity, which occurs only when |t0 | ≤ 2. It is possible, although quite
laborious, to prove this using knowledge of the Galois group and subfieldsof Q(β)(ζ2∞ )/Q(β). We omit the details.
We now discuss the case where m 6∈ {0, −1, −2} (we touch on m = −1
below). Theorem 4.1 and the strong form of the Hilbert Irreducibility Theorem imply that if m 6∈ {0, −1, −2} is fixed, then for all but a ‘thin’ set∼ Aut(Tn ) for f = (x − γ)2 + γ + m (see [14,of γ ∈ 1 Z we have Gn (f ) =2

section 6]). However, we would like to know something about G(f ) for suchspecializations. In order to emulate the proof of Theorem 4.1, one needs toknow arithmetic properties of the critical orbit {f n (γ) : n = 1, 2, . . .} of f ,specifically the assertion that the squarefree part of f n (γ) is divisible by aprime that does not divide f m (γ) for all m < n. We conjecture that thiscondition holds for all but finitely many n.
Conjecture 4.5 (Strong Dynamical Wieferich Prime Conjecture). Let b ∈1n
2 Z and f ∈ Z[x] be separable and quadratic such that {f (b) : n = 1, 2, . . .}is infinite. Then for all but finitely many n there exists a prime p withvp (f n (b)) odd and vp (f m (b)) = 0 for all m < n.
We first remark that Conjecture 4.5 implies that G(f ) has finite index
in Aut(T ) for all critically infinite quadratic f ∈ Z[x] all of whose iterates
are irreducible. This set of polynomials is quite a large subset of quadraticf (see [10, Theorem 3.5]). As for the name of Conjecture 4.5, recall thata Wieferich prime p is one satisfying 2p−1 ≡ 1 mod p2 . This condition isequivalent to the following: let an = 2n − 1, and let np be the smallest
index such that p | anp . Then p2 | anp . Currently only two Wieferichprimes are known, although even the statement that their complement isinfinite remains a conjecture (see e.g. [21]). Thus a reasonable analogue of
this conjecture in the dynamical setting would be that given an unboundedsequence {fn (b) : n = 1, 2, . . .}, there exist infinitely many p such thatvp (f n (b)) = 1 for some n but vp (f m (b)) = 0 for all m < n. Conjecture 4.5

THE IMAGE OF AN ARBOREAL GALOIS REPRESENTATION

11

represents a significant strengthening of this, albeit with only the stipulationthat vp (f n (b)) be odd.
Conjecture 4.5 can be proven in the case where the forward orbit of 0
{f n (0) : n = 1, 2, . . .} is finite but does not contain 0 [10, p. 19]. Thus for
instance the conjecture is true if f is of the form x2 + kx − k or x2 − kx − 1
for some k ∈ Z.
As for m = −1, matters are more mysterious, which perhaps is unsurprising given that the behavior in the generic case f = (x − t)2 + t − 1 ∈ Q(t)[x]is not known (see discussion on p. 8). We conjecture that for all but finitely
many specializations t0 ∈ Z, G(f ) has finite index in the group obtainedin the generic case. However, G(f ) has been not been explicitly computed
for any specializations t0 . The case t0 = −1, which gives f = (x + 1)2 − 2,
is particularly interesting because the splitting fields of the iterates f n are
2-extensions ramified only at 2 (and ∞). The suggestion has been madeby several people that the maximal such 2-extension should be the union ofthese splitting fields. This is conceivable since the Galois groups of the firstfew were computed (by Klüners and Fieker) and were large. Their orders(for n ≤ 7) were 2(2n+1 +1)/3

if n is even and 2(2n+1 +2)/3

if n is odd. If true

for general n, this would imply that G(f ) has Hausdorff dimension 2/3.
Further investigation leads to the following conjecture. With the standard generators a, b of the closure B of the Basilica group [1], let H be thesubgroup generated by [a, b] and aba, a normal subgroup with B/H infinite,
cyclic. Let Hn be the image of H in Aut(Tn ).
Conjecture 4.6. Let f = (x + 1)2 − 2. The Galois group of f n over Q(i)(a subgroup of index 2 in Gn (f )) is Hn for all n.
This implies that the union of the splitting fields of all the f n has Galois
group over Q(i) equal to the closure of H in Aut(T ). Using the explicit
presentation of B in [1], we compute that this closure is not a free pro-2
group (its class 5 quotient is smaller than that of the 2-generated free pro-2
group). On the other hand, Marksheitis [11] showed that the Galois groupover Q(i) of the maximal 2-extension unramified outside the prime above 2

12

NIGEL BOSTON AND RAFE JONES

is a free pro-2 group. Thus one consequence of Conjecture 4.6 is that G(f )is properly contained in the Galois group of the maximal 2-extension of Q
unramified outside {2, ∞} - in other words, this maximal 2-extension is not
generated by the roots of the iterates of f .
5. Conjectures
We gather the main conjectures made through this paper. They all pertain to the case where f ∈ Z[x] is quadratic. First we conjecture that every
irreducible quadratic f is settled. This implies that the images of Frobenius
elements under the corresponding arboreal Galois representation ρf are settled. Since Frobenius elements are dense in the image, we further conjecturethat the image G(f ) of ρf is a densely settled subgroup.
A subgroup commensurable to a densely settled subgroup is densely settled. There are many commensurability classes of densely settled subgroups
but we conjecture that few of these arise as possible G(f ) in the case wheref ∈ Z[x] is quadratic. In fact the only classes appear to be that of Aut(T )(in the case where f is critically infinite), that of the closure of the Basilica
group acting on T and a few subgroups of this group, that of the affinegroup on Z2 , that of Z2 , and that of the spherically homogeneous procyclic
subgroup. Conjectures 4.5 and 4.6 go in the direction of establishing thisclassification. It is worth remarking that if we allow f ∈ OK [x], where
K is an algebraic number field, then the classification should expand dueto the presence of additional conjugacy classes of critically finite quadratic
polynomials.

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