ORBIT COUNTING IN CONJUGACY CLASSES FOR FREE
GROUPS ACTING ON TREES
GEORGE KENISON AND RICHARD SHARP
Abstract. In this paper we study the action of the fundamental group of a
finite metric graph on its universal covering tree. We assume the graph is finite,
connected and the degree of each vertex is at least three. Further, we assume
an irrationality condition on the edge lengths. We obtain an asymptotic for
the number of elements in a fixed conjugacy class for which the associated
displacement of a given base vertex in the universal covering tree is at most T .
Under a mild extra assumption we also obtain a polynomial error term.
1. Introduction
Let G be a finite connected graph. We always assume that each vertex of G
has degree at least 3, in which case the fundamental group of G is a free group F
on k ≥ 2 generators. We make G into a metric graph by assigning to each edge e
a positive real length l(e). The length of a path in G is given by the sum of the
lengths of its edges. We assume the set of closed geodesics in G (i.e. closed paths
without backtracking) has lengths not contained in a discrete subgroup of R.
The universal cover of G is an infinite tree T and the metric on G lifts to a
metric dT on T . Let x ∈ F and fix a base vertex o ∈ T . We define L : F → R+
by L(x) = dT (o, ox). Let h denote the positive constant h = limT →∞ T1 log #{x ∈
F : L(x) ≤ T }. Guillopé [5] showed that
#{x ∈ F : L(x) ≤ T } ∼ cehT
as T → ∞,
fore some c > 0. Here f (T ) ∼ g(T ) means that f (T )/g(T ) → 1 as T → ∞.
Let C be a non-trivial conjugacy class in F . Then C is infinite and it is interesting
to study the restriction of the above counting problem to this conjugacy class, i.e.
to study the asymptotic behaviour of
NC (T ) := #{x ∈ C : L(x) ≤ T }.
The following is our main result.
Theorem 1.1. Suppose that G is a finite connected metric graph such that the
degree of each vertex is at least 3 and the set of lengths of closed geodesics in G is
not contained in a discrete subgroup of R. Let C be a non-trivial conjugacy class in
F . Then, for some constant C > 0, depending on C,
NC (T ) ∼ CehT /2 ,
as T → ∞.
We can also obtain a polynomial error term in our approximation to NC (T )
subject to a mild additional condition on the lengths of closed geodesics. This is
discussed in the final section (see Theorem 6.2).
In the case of a co-compact group of isometries of the hyperbolic plane, an
analogue of Theorem 1.1 was obtained by Huber [6] in the 1960s. If Γ is a cocompact Fuchsian group and C is a non-trivial conjugacy class, he showed that
#{g ∈ C : dH2 (o, og) ≤ T } ∼ C(Γ)eT /2 ,
as T → ∞,
for some C(Γ) > 0 (while the unrestricted counting function #{g ∈ Γ : dH2 (o, og) ≤
T } is asymptotic to a constant times eT ). Very recently, Parkkonen and Paulin
1
2
GEORGE KENISON AND RICHARD SHARP
[10] have studied the same problem in higher dimensions and variable curvature,
obtaining many results. In particular, they have shown that for the fundamental
group of a compact negatively curved manifold acting on its universal cover, the
conjugacy counting function has exponential growth rate equal to h/2, where h is the
topological entropy of the geodesic flow. They have an ergodic-geometric approach
using, in particular, the mixing properties of the Bowen-Margulis measure.
More closely related to our situation, suppose that G is a (q + 1)-regular graph
(i.e. each vertex has degree q + 1) with each edge given length 1. Then Douma [3]
showed that
NC (n) ∼ Cq b(n−l(C))/2c ,
as n → ∞
(for n ∈ Z+ ), for some C > 0, where l(C) is the length of the closed geodesic in the
conjugacy class C.
Since the first version of this paper was written, we learned that Broise-Alamichel,
Parkkonen and Paulin also have results for graphs and metric graphs which include
Theorem 1.1 and the result of Douma. They also consider the more general situation
of graphs of groups in the sense of Bass-Serre theory. We understand that an account
of this work is in preparation [1].
In contrast to the ergodic-geometric approach of Parkkonen and Paulin in [10] or
the use of spectral theory of the graph Laplacian in [3] (which is inspired by Huber’s
original spectral approach [6]), we use a method based on a symbolic coding of the
group F in terms of a subshift of finite type. We may then study a generating
function via the spectra of a family of matrices. In the next section, we set out the
background we shall need and describe how the lengths may be encoded in terms
of a function on our subshift. In section 3, we use this function to define a family of
matrices and in section 4 we use the preceding ideas to sketch a proof of Guillopé’s
result given above. In section 5 we introduce a generating function appropriate to
our problem and carry out an analysis which leads to the proof of Theorem 1.1. In
the final section we discuss error terms.
2. Preliminaries
A (finite, connected) graph G = (V, E) consists of a finite collection of vertices
V and edges E. Let E o denote the oriented edge set of the graph G. For each
e ∈ E o we indicate the edge with reversed orientation by e. A path is a sequence of
consecutive oriented edges e0 , . . . , en−1 ; and we call a path non-backtracking if, in
addition, ei+1 6= ei for i = 0, . . . , n − 1. Path e0 , . . . , en−1 is said to be closed if the
terminal vertex of en−1 and the initial vertex of e0 are the same. We say a closed
path e0 , . . . , en−1 is a closed geodesic if the path is non-backtracking and en−1 6= e0 ,
i.e. each path given by a cyclic permutation of e0 , . . . , en−1 is non-backtracking.
The condition that each vertex has degree at least 3 ensures that the fundamental
group of G is a free group F on k ≥ 2 generators and that the universal cover is an
infinite tree T . We put a metric on G by assigning a positive length to each edge
and this lifts to a metric on T .
Fix a generating set A = {a1 , . . . , ak }, k ≥ 2, for the free group F and write
−1
−1
A−1 = {a−1
, is a
1 , . . . , ak }. We shall say x0 · · · xn−1 , with each xi ∈ A ∪ A
−1
reduced word if xi+1 6= xi for i = 0, . . . , n − 1 and a cylically reduced word if, in
addition, xn−1 6= x−1
0 . Each non-identity element of F has a unique representation
as a reduced word. For x ∈ F the word length |x| of x is the number of terms in its
reduced word representation. Let C be a non-trivial conjugacy class in F . Then C
contains a cyclically reduced word x0 · · · xn−1 in A ∪ A−1 . The only other cyclically
reduced words in this conjugacy class are obtained by cyclic permutation and the
elements of C represented by non-cyclically reduced words have word length greater
than n.
ORBIT COUNTING IN CONJUGACY CLASSES
3
To study our counting problem, we will introduce a symbolic dynamical system
called a subshift of finite type. Before specialising to the particular system we will
use, we make some general definitions. Let S be a finite set of symbols and let A
be a |S| × |S| matrix with entries in {0, 1}. A finite string of symbols s1 , . . . , sn in
S is called an admissible word if A(si , si+1 ) = 1 for i = 1, . . . , n − 1. We define Σ+
A
to be the set of all infinite admissible words:
o
n
∞
Z+
Σ+
: A(xn , xn+1 ) = 1 ∀n ∈ Z+ .
A = (xn )n=0 ∈ S
We say that A is irreducible if, for each (s, s0 ) ∈ S × S, there exists n ≥ 1 such
that An (s, s0 ) > 0 and that A is aperiodic if there exists n ≥ 1 such that, for each
+
(s, s0 ) ∈ S × S, An (s, s0 ) > 0. Writing x = (xn )∞
n=1 ∈ ΣA , for a fixed 0 < θ < 1
+
we endow ΣA with the metric defined by d(x, x) = 0 and, for x 6= y, d(x, y) = θk ,
where k = min{n ∈ N : xn 6= yn }. That A is irreducible is sufficient to ensure this
+
metric makes Σ+
A into a Cantor set. The dynamics on ΣA is given by the shift map
+
σ : Σ+
A → ΣA defined by (σx)n = xn+1 .
A function f : Σ+
A → R is said to be locally constant if there exists an N ∈ N
such that for any two elements x, y ∈ Σ+
A with xn = yn for every 0 ≤ n ≤ N we have
f (x) = f (y). If f is locally constant then it is Hölder continuous (i.e. there exist
positive constants α and κ such that |f (x) − f (y)| ≤ κd(x, y)α , for every x, y ∈ Σ+
A ).
We will use the notation
f n = f + f ◦ σ + · · · + f ◦ σ n−1 .
Two Hölder continuous functions f, g : Σ+
A → R are said to be cohomologous if
f = g + u ◦ σ − u for some continuous function u : Σ+
A → R. We characterise the
cohomology of Hölder functions f and g in terms of the periodic points of σ: f and
g are cohomologous if and only if f n (x) = g n (x) for every x ∈ Fixn (σ) := {x ∈
n
Σ+
A : σ x = x} ([12], Proposition 3.7).
Let M denote the the set of σ-invariant probability measures on Σ+
A . We denote
by h(µ) the measure theoretic entropy of σ with respect to µ. For a continuous
function f : Σ+
A → R, we define its pressure P (f ) by
Z
P (f ) = sup h(µ) + f dµ .
µ∈M
We say that m ∈ M is an equilibrium state for f if the supremum is attained at m.
When f is a Hölder continuous function the Variational Principle ([12], Theorem
3.5) tells us the equilibrium state is unique.
We will use the following standard result on non-negative aperiodic matrices.
(Here, we do not assume the matrix only has entries in {0, 1}.)
Theorem 2.1 (Perron-Frobenius Theorem [4]). Suppose that A is an aperiodic
matrix with non-negative entries. Then A has a simple and positive eigenvalue β
such that β is strictly greater in modulus than all the remaining eigenvalues of A.
The left and right eigenvectors associated to the eigenvalue β have strictly positive
entries. Moreover, β is the only eigenvalue of A that has an eigenvector whose
entries are all non-negative.
We will also need to consider the spectra of complex matrices and will use the
following theorem.
Theorem 2.2 (Wielandt’s Theorem [4]). Suppose that A is a square matrix with
complex entries and let |A| be the matrix whose entries are given by |A|(s, s0 ) =
|A(s, s0 )|. Suppose further that |A| is aperiodic and let β be its maximal eigenvalue
guaranteed by the Perron-Frobenius Theorem. Then the moduli of the eigenvalues
of A are bounded above by β. Moreover, A has an eigenvalue of the form βeiθ (with
θ ∈ [0, 2π]) if and only if A = eiθ D|A|D−1 where D is a diagonal matrix whose
entries along the main diagonal all have modulus one.
4
GEORGE KENISON AND RICHARD SHARP
We now introduce a specific subshift of finite type suitable for our analysis. We
define a 2k × 2k matrix A, with rows and columns indexed by the set S = A ∪ A−1 ,
by A(a, b) = 1 if b is not the inverse of a, and A(a, b) = 0 otherwise. It is clear that
A is a aperiodic. We will write Σ+ = Σ+
A . (It will be convenient to maintain an
ambiguity between reduced words in A ∪ A−1 as elements of F and as admissible
words for the shift Σ+ .)
Remark 2.3. Another subshift of finite type naturally associated to the graph
is obtained by taking the symbols S to be the set of oriented edges and setting
A(e, e0 ) = 1 if and only if e0 follows e in the graph. The advantage of our approach is
that is makes it easier to systematically enumerate the elements of a given conjugacy
class. On the other hand, it requires more work to represent the edge lengths and
we introduce a function that does this below.
In order to represent elements of F in this framework, we introduce an augmented
sequence space that associates to each group element in F an infinite sequence.
Introduce a “dummy symbol” 0 and define a (2k + 1) × (2k + 1) matrix B, indexed
by A ∪ A−1 ∪ {0}, by

−1

A(a, b) if a, b ∈ A ∪ A
−1
B(a, b) = 1
if a ∈ A ∪ A ∪ {0} and b = 0


0
if a = 0 and b ∈ A ∪ A−1
+
+
and write Σ+
0 = ΣB . In particular, we note that, apart from 0̇ ∈ Σ0 , where 0̇
+
+
+
denotes an infinite sequence of 0s, the shift maps σ : Σ → Σ and σ : Σ+
0 → Σ0
have the same periodic points. We associate to a group element x ∈ F , with reduced
word representation x0 · · · xn−1 , the infinite sequence (x0 , . . . , xn−1 , 0̇).
As in [15] and [16], we define a function on Σ+
0 from which one can recover
L : F → R+ (where we recall that L(x) = dT (o, ox) for some fixed base point
o ∈ T ). Suppose that x ∈ Σ+
0 is a sequence of the form (x0 , . . . , xn−1 , 0̇). We let
r(x0 , x1 , . . . , xn−1 , 0̇) = L(x0 x1 · · · xn−1 ) − L(x1 · · · xn−1 ).
We use the following lemma from [16] to extend this definition to all sequences in
Σ+
0.
Lemma 2.4. There exists N ∈ N such that if n ≥ N and x0 · · · xn−1 is a reduced
word then
L(x0 x1 · · · xn−1 ) − L(x1 · · · xn−1 ) = L(x0 x1 · · · xN −1 ) − L(x1 · · · xN −1 ).
Definition 2.5. We define r : Σ+
0 → R by r(0̇) = 0,
r((xn )∞
n=0 ) = L(x0 x1 · · · xn−1 ) − L(x1 · · · xn−1 )
if (xn )∞
n=0 = (x0 , x1 , . . . , xn−1 , 0̇) with n < N , and
r((xn )∞
n=0 ) = L(x0 x1 · · · xN −1 ) − L(x1 · · · xN −1 )
otherwise.
If min{k ∈ N : xk 6= yk } > N then r(x) = r(y), so r : Σ+
0 → R is locally constant.
Furthermore, the next lemma follows from the definition (cf. [16]).
Lemma 2.6. Suppose x0 x1 · · · xn−1 is a reduced word. Then
L(x0 x1 · · · xn−1 ) = rn (x0 , x1 , . . . , xn−1 , 0̇).
The periodic orbit sums of rn (x) are related to the lengths of the closed geodesics
L(x0 · · · xn−1 ) in the following lemma.
Lemma 2.7. Let γ be the unique closed geodesic corresponding to the periodic orbit
{x, σx, . . . , σ n−1 x} (σ n x = x). Then rn (x) = l(γ). In particular, {rn (x) : x ∈
Fixn , n ≥ 1} is not contained in a discrete subgroup of R.
ORBIT COUNTING IN CONJUGACY CLASSES
5
Proof. Let (x0 , . . . , xn−1 )(m) denote the m-fold concatenation of the cyclically reduced word (x0 , . . . , xn−1 ). For sufficiently large m, we have
|rmn (x) − rmn ((x0 , . . . , xn−1 )(m) , 0̇)| ≤ 2N krk∞
and so, rmn (x) = mrn (x) and rmn ((x0 , . . . , xn−1 )(m) , 0̇) = L((x0 · · · xn−1 )m ), we
deduce
1
L((x0 · · · xn−1 )m ).
rn (x) = lim
m→∞ m
Now consider the closed geodesic γ in G. This lifts to a geodesic path in T , from
some vertex p to pg, where g ∈ F is conjugate to x0 · · · xn−1 . For each m ≥ 1, we
have ml(γ) = dT (p, pg m ). For any vertex q ∈ T , the triangle inequality gives
dT (p, pg m ) − 2dT (p, q) ≤ dT (q, qg m ) ≤ dT (p, pg m ) + 2dT (p, q),
so that
1
dT (q, qg m ).
m→∞ m
= w−1 gw then putting q = ow−1 gives
l(γ) = lim
If x0 · · · xn−1
1
1
dT (ow−1 , ow−1 g m ) = lim
dT (o, o(x0 · · · xn−1 )m )
m→∞ m
m
1
L((x0 · · · xn−1 )m ).
= lim
m→∞ m
l(γ) = lim
m→∞
This completes the proof.
3. Matrices and spectra
It will be convenient to use a family of matrices which encode information about
the edge lengths. We can only do this directly if N = 2, i.e. if r(x) = r(x0 , x1 ).
e + by state splitting so
More generally, we introduce a new subshift of finite type Σ
that the new symbol set consists of the set WN −1 of admissible words of length
e defined by
N − 1 in A ∪ A−1 and transition matrix A
(
1 if y0 = x1 , . . . , yN −3 = xN −2
e
A((x0 , . . . , xN −2 ), (y0 , . . . , yN −2 )) =
0 otherwise.
e is aperiodic and the shift σ̃ : Σ
e+ → Σ
e + is topologically conjugate to
Then A
e + → R by setting
σ : Σ+ → Σ+ ([9], Chapter 2). We define a function r̃ : Σ
r̃((x0 , . . . , xN −2 ), (x1 , . . . , xN −1 )) = r((xn )∞
n=0 ).
es for s ∈ C by setting, for x, y ∈ WN −1 ,
We now define a family of matrices A
es (x, y) = e−sr̃(x,y) A(x,
e y).
A
es is aperiodic and so, by the Perron-Frobenius Theorem,
When s ∈ R, the matrix A
has a simple and positive eigenvalue β(s), which is strictly greater in modulus than
es . Let u(s) and v(s) denote the left and right
all of the other eigenvalues of A
e
eigenvalues of As corresponding to β(s). We normalise the eigenvectors so that
X
(u(s))x = 1; and
x∈WN −1
X
(u(s))x (v(s))x = 1.
x∈WN −1
We have the following lemma.
6
GEORGE KENISON AND RICHARD SHARP
es . Then
Lemma 3.1. Suppose that β(s) is the maximal eigenvalue of the matrix A
β(s) is real analytic and
Z
0
β (s) = −β(s) r̃ dµ−sr̃ ,
R
where µ−sr̃ is the equilibrium state for −sr̃. Furthermore, r̃ dµ−sr̃ > 0.
e + → Σ+ be the
Proof. The first two statements are standard, see [12]. Let ϕ : Σ
obvious topological conjugacy. Then r̃ = r ◦ ϕ and it follows easily from uniqueness
of
R equilibriumR states that ϕ∗ µ−sr̃ = µ−sr , the equilibrium state−1ofn −sr. Thus
r̃ dµ−sr̃ = r dµ−sr . For any n ≥ 1, r is cohomologous to n r and, for n
sufficiently large, rn is strictly positive. These observations prove positivity of the
integrals.
Corollary 3.2. There exists a unique positive real number h such that β(h) = 1.
ens , one can easily check that β(0) > 1 and
Proof. By comparing with the trace of A
that β(s) < 1 for sufficiently large s. By Lemma 3.1, β(s) is strictly decreasing, so
the required number h exists and is unique.
We shall see in the next section that this number h agrees with the growth rate
h introduced in section 1.
es for s ∈ C. In particular, we have the following
We will also need to consider A
result.
eh+it does not have 1 as an eigenvalue.
Lemma 3.3. For t 6= 0, the matrix A
Proof. By construction, {r̃n (x) : x ∈ Fixn (σ̃), n ≥ 1} = {rn (x) : x ∈ Fixn (σ), n ≥
1} and, by Lemma 2.7, this is not contained in a discrete subgroup of R. It then
follows from [11] that the result holds.
es , indexed by the
Finally in this section, we introduce a larger family of matrices B
words in WN −1 together with strings of length N −1 of the form (x0 , . . . , xq , 0, . . . , 0),
where (x0 , . . . , xq ) ∈ Wq+1 for 1 ≤ q ≤ N − 2 and the string (0, . . . , 0), and defined
by

e

As (x, y) if x, y ∈ WN −1
es (x, y) = e−sr̃(x,y) if x = (x0 , . . . , xq , 0, . . . , 0), y = (x1 , . . . , xq , 0, 0, . . . , 0)
B


0
otherwise.
e0 should not be thought of as a transition matrix owing to the term
(Note that B
e0 ((0, . . . , 0), (0, . . . , 0)) = 0.) It is easy to see that B
es has the same non-zero
B
es (though B
es has additional zero-valued eigenvalues).
spectrum as A
For s ∈ R, let û(s) and v̂(s) denote, respectively, the left and right eigenvectors
es associated to the eigenvalue β(s). We denote by Wn(m) the strings of length
of B
n with m non-zero terms:
Wn(m) = {(x0 , . . . , xm−1 , 0, . . . , 0) : (x0 , . . . , xm−1 ) ∈ Wm }.
The entries of û(s) are given as follows. We first define a vector û0 and then
choose a normalisation to give û. To start, we let (û0 (s))y = (u(s))y if y ∈ WN −1 .
(N −2)
Then, for y ∈ WN −1 , the entry (û0 (s))y is given by
X
1
es (x, y).
(û0 (s))y =
(û0 (s))x B
β(s)
x∈WN −1
(q)
We continue our construction iteratively: for y ∈ WN −1 (with q ≤ N − 2) we set
X
1
es (x, y).
(û0 (s))y =
(û0 (s))x B
β(s)
(q+1)
x∈WN −1
ORBIT COUNTING IN CONJUGACY CLASSES
7
Clearly, each entry of û0 (s) is positive by construction. Finally, we define û(s) =
((û0 (s))(0,...,0) )−1 û0 (s) (so that (û(s))(0,...,0) = 1).
The right eigenvector v̂(s) is given by v̂(s) = ((û0 (s))(0,...,0) )v̂0 (s), where v̂0 (s)
has entries given by
(
(v(s))y if y ∈ WN −1
(v̂0 (s))y =
0
otherwise.
Thus v̂(s) is normalised so that û(s) · v̂(s) = 1.
4. Guillopé’s Theorem
In this section we explain why the number h defined above by β(h) = 1 is equal
to the growth rate h = limT →∞ T1 log #{x ∈ F : L(x) ≤ T } from the introduction.
We will do this by sketching a proof of Guillopé’s theorem. This will also serve as
a prelude to the analysis in the next section, where the arguments will be given in
more detail.
To evaluate the asymptotic behaviour of #{x ∈ F : L(x) ≤ T } we first establish
analytic properties of the complex generating function
η(s) =
∞ X
X
∞
X
e−sL(x) =
X
e−sL(x) + φ(s)
n=N −1 |x|=n
n=1 |x|=n
where φ(s) is an entire function. Letting 1 = (1, . . . , 1), we can rewrite η(s) in
es :
terms of the matrices A
η(s) =
∞
X
ens 1T + φ(s).
1A
n=1
This converges absolutely for β(Re(s)) < 1, i.e. for Re(s) > h.
Since β(s) is a simple eigenvalue, it varies analytically and we can show that
η(s) = c0 /(s − h) + φ1 (s) where φ1 (s) is an analytic function in a neighbourhood
of s = h and c0 = −(v(s) · 1)/β 0 (h) > 0. Furthermore, using Lemma 3.3, we may
show that η(s) has no further poles on the line Re(s) = h.
The generating function η(s) is related to the counting function N (T ) := #{x ∈
F : L(x) ≤ T } by the following Stieltjes integral:
Z ∞
η(s) =
e−sT dN (T ).
0
This enables us to apply the Ikehara-Wiener Tauberian Theorem.
Theorem 4.1 (Ikehara-Wiener
R ∞ Tauberian Theorem ([12], Theorem 6.7)). Suppose
that the function η(s) = 0 e−sT dN (T ) is analytic for Re(s) > h, has a simple
pole at s = h, and the function η(s) − c0 /(s − h) has an analytic extension to a
neighbourhood of Re(s) ≥ h. Then
N (T ) ∼
c0 ehT
,
h
as T → ∞.
As a consequence, we recover Guillopé’s result that
#{x ∈ F : L(x) ≤ T } ∼ cehT
as T → ∞,
for some c > 0, and hence the formula h = limT →∞
1
T
log #{x ∈ F : L(x) ≤ T }.
8
GEORGE KENISON AND RICHARD SHARP
5. A complex generating function
Suppose that C is a non-trivial conjugacy class in F and let
k = min |x|,
x∈C
i.e. the minimum word length of an element of C. For m ≥ k, write Cm = {x ∈
C : |x| = m}. Then each g ∈ Ck is represented by a cyclically reduced word g1 · · · gk
and the other elements of Ck are obtained from g by cyclic permutation. For a given
g ∈ Ck , we denote by Wm (g) the elements of F whose reduced word representation
w1 · · · wm is such that w1 6= g1 , gk−1 . We observe that, for w = w1 · · · wm ∈ Wm , the
word w−1 gw has no pairwise cancellation of terms if w1 6= g1 , gk−1 , i.e. if w ∈ Wm (g).
Provided m ≥ N − 1, Wm (g) can be written as a disjoint union
[
Wm (g) =
Wm (g, y),
y∈WN −1 (g)
where y = y1 · · · yN −1 and Wm (g, y) = {w ∈ Wm (g) : w = y1 · · · yN −1 wN · · · wm }.
It is clear that, for all n ≥ 0, Ck+n is non-empty if and only if n is even. Each
element x ∈ Ck+2m has a unique reduced word representation of the form w−1 gw =
−1
wm
· · · w1−1 g1 · · · gk w1 · · · wm , for some g ∈ Ck , y ∈ WN −1 (g) and w ∈ Wm (g, y).
In order to study the counting problem in the conjugacy class C, we define a
generating function
ηC (s) =
X
e−sL(x) =
∞
X
X
e−sL(x) .
m=1 x∈Ck+2m
x∈C
We write ηC (s) in terms of Ck , WN −1 (g) and Wm (g, y) as follows:
X
ηC (s) =
X
∞
X
X
e−sL(w
−1
gw)
+ φ1 (s),
g∈Ck y∈WN −1 (g) m=N −1 w∈Wm (g,y)
PN −2 P
where φ1 (s) is the entire function given by φ1 (s) = m=1 x∈Ck+2m e−sL(x) .
The expression for L(w−1 gw) in the next lemma follows easily from L(w−1 gw) =
k+2m
r
(w−1 gw, 0̇).
Lemma 5.1. Suppose that y ∈ WN −1 (g) and w ∈ Wm (g, y). Then
L(w−1 gw) = 2L(w) + L(y −1 gy) − 2L(y).
Using Lemma 5.1, we obtain that, modulo the addition of the entire function
φ1 (s),
ηC (s) =
X
X
e−s(L(y
−1
gy)−2L(y))
g∈Ck y∈WN −1 (g)
∞
X
X
e−2sL(w) .
m=N −1 w∈Wm (g,y)
Since w ∈ Wm (g, y) and L(w) = rm (w, 0̇), we have
X
em
e−2sL(w) = eT
y B2s e(0,...,0) ,
w∈Wm (g,y)
where ey and e(0,...,0) denote standard unit column vectors. Thus ηC (s) can be
written as
∞
X
X
X
−s(L(y −1 gy)−2L(y))
em
ηC (s) =
e
eT
y B2s e(0,...,0) .
g∈Ck y∈WN −1 (g)
again modulo the addition of an entire function.
We will need the following simple lemma.
m=N −1
ORBIT COUNTING IN CONJUGACY CLASSES
9
Lemma 5.2. Suppose that β is a simple eigenvalue of a matrix M and that u and v
are associated left and right eigenvectors, chosen so that u·v = 1. Then an arbitrary
vector w has a unique expression as w = cv + ṽ, where ṽ lies in the span of the
generalised eigenspaces associated to the other eigenvalues of M and c = u · w.
Proof. Suppose that w = cv + ṽ, for some c ∈ C. Then
u · w = u · (cv + ṽ) = c(u · v) + u · ṽ = c + u · ṽ.
Now u is orthogonal to any generalised right eigenvector of M associated to an
eigenvalue other than β, so that u and ṽ are orthogonal. Thus c = u · w.
Proposition 5.3. The generating function ηC (s) is analytic for Re(s) > h/2, has
a simple pole at s = h/2 with positive residue and, apart from this, has an analytic
extension to a neighbourhood of Re(s) ≥ h/2.
e2s has spectral radius β(2s). Since this is strictly decreasing
Proof. For s ∈ R, B
P∞
em
and β(h) = 1, it is clear that m=N −1 eT
y B2s e(0,...,0) converges for s > h/2 and
hence that, for s ∈ C, ηC (s) is analytic for Re(s) > h/2.
Now suppose s0 = σ + it ∈ C. By Wielandt’s Theorem (Theorem 2.2) and the
es and B
es , either
correspondence between the spectra of A
es ) = spr(A
eσ ), in which case B
es has a simple eigenvalue β(s0 ) with
(1) spr(B
0
0
|β(s0 )| = β(σ) and such that the remaining eigenvalues are strictly smaller
in modulus; or
es ) < spr(B
eσ ).
(2) spr(B
0
When (1) holds, standard eigenvalue perturbation theory gives that the simple
eigenvalue β(s) persists and is analytic for s in a neighbourhood of s0 , as are the
corresponding left and right eigenvectors û(s) and v̂(s) [8]. By Lemma 5.2 and
recalling that (û(s))(0,...,0) = 1, we have e(0,...,0) = v̂(2s) + ṽ(2s), where ṽ(2s) is a
vector in the subspace spanned by the eigenvectors associated to the non-maximal
e2s . Thus we have
eigenvalues of B
∞
∞
∞
X
X
X
m
m
e
em
eT
B
e
=
(e
·
v̂(2s))β(2s)
+
eT
y
y 2s (0,...,0)
y B2s ṽ(2s)
m=N −1
m=N −1
=
(ey · v̂(2s))β(2s)
1 − β(2s)
m=N −1
N −1
+ φ2 (s),
where φ2 (s) is analytic in a neighbourhood of s0 . Therefore, ηC (s) is analytic in
a neighbourhood of s0 unless β(2s0 ) = 1. For Re(s0 ) = h/2, Lemma 3.3 tells us
that this only occurs when s0 = h/2. When (2) holds, we immediately obtain that
P∞
T em
m=N −1 ev B2s e(0,...,0) converges and hence that ηC (s) converges to an analytic
function for s in a neighbourhood of s0 .
For s in a neighbourhood of h/2, we have that, modulo an analytic function,
X
X
−1
(ey · v̂(2s))β(2s)N −1
ηC (s) =
e−s(L(y gy)−2L(y))
.
1 − β(2s)
g∈Ck y∈WN −1 (g)
From the analyticity of β and the fact that β(h) = 1, we obtain that, in a neighbourhood of h/2,
c
ηC (s) =
+ φ3 (s),
s − h/2
where
S φ3 (s) is analytic and c > 0. The latter holds because ey · v̂(h) > 0, for each
y ∈ g∈Ck WN −1 (g), and, by Lemma 3.1, −1/(2β 0 (h)) > 0.
Combining the above observations, we have that ηC (s) is analytic for Re(s) >
h/2 and, apart from a simple pole at s = h/2, has an analytic extension to a
neighbourhood of Re(s) ≥ h/2. Furthermore, the residue at the simple pole is
positive.
10
GEORGE KENISON AND RICHARD SHARP
Since ηC (s) is related to the counting function NC (T ) = #{x ∈ C : dT (o, ox) ≤ T }
via the Stieltjes integral
Z ∞
e−sT dNC (T ),
ηC (s) =
0
we may apply the Ikehara-Wiener Tauberian Theorem (Theorem 4.1) to conclude
that
ehT /2
NC (T ) ∼ C
,
h/2
for some C > 0. This completes the proof of Theorem 1.1.
6. Error terms
In this final section we discuss the error terms which may appear when estimating
NC (T ). We first note the following, which may be deduced from the analysis above
and the arguments in [13] (Propositions 6 and 7).
Proposition 6.1. There is never an exponential error term in Theorem 1.1, i.e.
for no > 0 do we have NC (T ) = CehT /2 + O(e(h−)T /2 ).
A more interesting problem is to ask when there is a polynomial error term for
NC (T ). Recall that an irrational number α is said to be Diophantine if there exist
c > 0 and β > 1 such that |qα − p| ≥ cq −β for all p, q ∈ Z, q > 0. We have the
following.
Theorem 6.2. Suppose that G is a finite connected metric graph such that the
degree of each vertex is at least 3. Suppose also that G contains two closed geodesics
γ and γ 0 such that l(γ)/l(γ 0 ) is Diophantine. Then there exists δ > 0 such that
NC (T ) = CehT /2 + O(ehT /2 T −δ ).
Note that, since the lengths of the closed geodesics in G are not contained in
a discrete subgroup of R if and only if there are two closed geodesics the ratio of
whose lengths is irrational, the hypothesis of Theorem 6.2 is strictly stronger that
that of Theorem 1.1.
For the remainder of this section, it will be convenient to write s = σ + it.
es , follows
The next proposition, which gives a bound on powers of the matrix B
from the work of Dolgopyat [2] on transfer operators where it is a key ingredient in
studying the mixing rate of flows (cf. also [14]).
Proposition 6.3. Under the hypotheses of Theorem 6.2, there exist constants
τ, C1 , C2 > 0 and t1 ≥ 1 such that when |t| ≥ t1 and m ≥ 1,
m−1
2N m
eσ+it
kB
k ≤ C1 |t|β(σ)2N m 1 − |t|−τ
,
where N = bC2 log |t|c.
The elementary inequality
l
eσ+it
kB
k ≤ β(σ)n D1 |t| + θl D2 ,
for some constants D1 , D2 > 0 and all l > 0 (cf. Proposition 2.1 of [12]), and the
en
previous
n result provide a bound on kBs k for all n ∈ N. Let n = 2N m + l where
m = 2N and l ∈ {0, . . . , 2N − 1}, then, for |t| ≥ t1
n
eσ+it
kB
k ≤ C3 |t|2 β(σ)n (1 − |t|−τ )m−1 ,
for some C3 > 0 We use this to study the analyticity of ηC (s) (using a simpler
version of the arguments in [14]).
ORBIT COUNTING IN CONJUGACY CLASSES
11
Proposition 6.4. There exist constants ρ > 0 and t2 ≥ t1 such that ηC (s) has an
analytic extension to the region
R(ρ) = {σ + it ∈ C : 2σ > h − 1/|2t|ρ , |t| ≥ t2 },
where it satisfies the bound |ηC (σ + it)| = O(|t|2+ρ ).
Proof. Suppose that σ + it ∈ C satisfies 2σ > h − |2t|−ρ and |2t| ≥ t1 , with ρ > τ .
Since
β(2σ) = 1 + β 0 (h)(2σ − h) + O((2σ − h)2 )
with β 0 (h) < 0, for |t| ≥ t2 , where t2 ≥ t1 , we have (1 − |2t|−τ )1/2N < β(2σ)−1 .
P∞
e m e(0,...,0) |, for some D > 0. Thus, for
We can estimate |ηC (s)| ≤ D m=1 |B
2s
σ + it ∈ R(ρ), we have
∞
∞
X
X
bm/2N c−1
em
e
|
≤
|B
C3 |2t|2 β(2σ) 1 − |2t|−τ
2(σ+it) (0,...,0)
m=1
m=1
=
C3 |2t|2 β(2σ)
2−2/N
1/2N
(1 − |2t|−τ )
(1 − β(2σ)(1 − |2t|−τ )
2
C4 |2t|
≤
= O(|t|2+ρ ),
1/2N
(1 − β(2σ)(1 − |2t|−τ )
)
)
which shows that ηC (s) is analytic in the desired region and gives the bound.
Let ξC (s) be the normalised generating function given by
X
ξC (s) = ηC (sh/2) =
e−shL(x)/2 .
x∈C
We immediately deduce that ξC (s) is analytic in the half-plane Re(s) > 1, has
an analytic extension to a neighbourhood of Re(s) ≥ 1 save for the simple pole
at s = 1, which has positive residue, and, furthermore, that there exist positive
constants ρ and t3 = 2t2 /h such that ξC (s) has an analytic extension to
1
Rξ (ρ) = σ + it : σ > 1 − ρ+1 ρ , |t| > t3 ; and
h |t|
where it satisfies |ξC (s)| = O(|t|2+ρ ).
Let us introduce a normalised counting function
X
ψ0 (T ) =
1.
ehL(x)/2 ≤T
Adapting the arguments of [14], we will establish an error term for ψ0 (T ), from
which Theorem 6.2 will follow since ψ0 (ehT /2 ) = NC (T ). We introduce the following
RT
family of auxiliary functions. Let ψ1 (T ) = 1 ψ0 (u) du and continue inductively so
that
Z T
X
1
(T − ehL(x)/2 )k .
ψk (T ) =
ψk−1 (u) du =
k! hL(x)/2
1
e
≤T
We use the following identity ([7], Theorem B, page 31) to connect the functions
ξC (s) and ψk (T ). If k is a positive integer and d > 0 we have
(
Z d+i∞
0
if 0 < y < 1,
1
ys
ds = 1
k
2πi d−i∞ s(s + 1) · · · (s + k)
if y ≥ 1.
k! (1 − 1/y)
This gives us the following.
Lemma 6.5. For d > 1 we may write
Z d+i∞
1
ξC (s)T s+k
ψk (T ) =
ds.
2πi d−i∞ s(s + 1) · · · (s + k)
12
GEORGE KENISON AND RICHARD SHARP
We briefly outline the method to approximate the integral for ψk (T ). First,
compare the integral for ψk (T ) to the truncated integral on the line segment [d −
iR, d + iR] where R = (log T )ε and 0 < ε < 1/ρ. Let us choose d = 1 + 1/ log T and
then since ξC (d) = O((d − 1)−1 ), we deduce
Z d+iR
1
ξC (s)T s+k
(log T )T k+1
T
=O
.
ds = O
ψk (T ) −
2πi d−iR s(s + 1) · · · (s + k) Rk+1
(log T )kε
We evaluate the truncated integral using Cauchy’s Residue Theorem. Consider
a closed contour Γ ∪ [d − iR, d + iR]. Here Γ is the union of the line segments
[d + iR, c(R) + iR], [c(R) − iR, d − iR] and [c(R) + iR, c(R) − iR], where c(R) =
1 − 1/(2hρ+1 Rρ ) so that Γ lies in Rξ (ρ). Note that Γ ∪ [d − iR, d + iR] encloses
the simple pole of ξC (s) at s = 1. Cauchy’s Residue Theorem gives that, for some
c > 0,
Z d+iR
Z
ξC (s)T s+k
cT k+1
1
ξC (s)T s+k
1
ds =
+
ds.
2πi d−iR s(s + 1) · · · (s + k)
(k + 1)! 2πi Γ s(s + 1) · · · (s + k)
We consider the contribution made by each of the line segments in Γ. First,
integrating over the interval [d + iR, c(R) + iR] we have
Z
T d+k Z c(R)+iR
c(R)+iR
T d+k
ξC (s)T s+k
ξC (s)ds = O
ds ≤
d+iR
s(s + 1) · · · (s + k) Rk+1 d+iR
(log T )(k−ρ−1)ε
and similarly for [c(R) − iR, d − iR],
Z
d−iR
T d+k
ξC (s)T s+k
.
ds = O
c(R)−iR s(s + 1) · · · (s + k) (log T )(k−ρ−1)ε
We estimate the modulus of the integral along [c(R) + iR, c(R) − iR] by
Z
!
Z R
c(R)−iR
ξC (s)
c(R)+k
c(R)+k 1+ρ−k
T
ds = O T
t
dt
c(R)+iR s(s + 1) · · · (s + k) 1
= O(T c(R)+k R2+ρ−k ),
which means for any positive γ we have
T c(R)+k R2−k = T k+1 e
−
log T
2hρ+1 (log T )ερ
(log T )(2+ρ−k)ε = O(T k+1 (log T )−γ ).
Together the integral estimates give us an error term for ψk (T ):
T k+1
0 k+1
ψk (T ) = c T
+O
(log T )(k−ρ−1)ε
where c0 > 0. Then repeatedly applying the inequality
ψj−1 (T − ∆T )∆T ≤ ψj (T ) − ψj (T − ∆T ) ≤ ψj−1 (T )∆T,
j−k−1
ε
where ∆T = T (log T )−(k−ρ−1)2
, we obtain ψ0 (T ) = CT + O T (log T )−δ
where C, δ > 0. Thus we have the error term NC (T ) = CehT /2 + O(ehT /2 T −δ ),
completing the proof of Theorem 6.2.
References
[1] A. Broise-Alamichel, J. Parkkonen and F. Paulin, Counting paths in graphs, in preparation.
[2] D. Dolgopyat, Prevalence of rapid mixing in hyperbolic flows, Ergodic Theory Dynam. Systems 18, 1097-1114, 1998.
[3] F. Douma, A lattice point problem on the regular tree, Discrete Math. 311, 276-281, 2011.
[4] F. Gantmacher, The Theory of Matrices, Volume II, AMS Chelsea Publishing, Providence
RI, 2000.
[5] L. Guillopé, Entropies et spectres, Osaka J. Math. 31, 247-289, 1994.
[6] H. Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen II,
Math. Ann. 142, 385-398, 1961.
ORBIT COUNTING IN CONJUGACY CLASSES
13
[7] A. Ingham, The distribution of prime numbers, Cambridge Mathematical Library, Cambridge
University Press, Cambridge, 1990.
[8] T. Kato, Perturbation theory for linear operators, Reprint of the 1980 edition, Classics in
Mathematics, Springer-Verlag, Berlin, 1995.
[9] B. Kitchens, Symbolic Dynamics: One-Sided, Two-Sided and Countable State Markov Shifts,
Universitext, Springer-Verlag, Berlin Heidelberg, 1998.
[10] J. Parkkonen and F. Paulin, On the hyperbolic orbital counting problem in conjugacy classes,
Math. Z. 279, 1175-1196, 2015.
[11] W. Parry, An analogue of the prime number theorem for closed orbits of shifts of finite type
and their suspensions, Israel J. Math. 45, 41-52, 1983.
[12] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic
dynamics, Astérisque 187-188, Société Mathematique de France, 1990.
[13] M. Pollicott, Meromorphic extensions of generalised zeta functions, Invent. math. 85, 147-164,
1986.
[14] M. Pollicott and R. Sharp, Error terms for closed orbits of hyperbolic flows, Ergodic Theory
Dynam. Systems 21, 545-562, 2001.
[15] R. Sharp, Distortion and entropy for automorphisms of free groups, Discrete Contin. Dynam.
Systems 26, 347-363, 2009.
[16] R. Sharp, Comparing length functions on free groups, in “Spectrum and Dynamics”, CRM
Conference Proceedings & Lecture Notes 52, 185-207, 2010.
Mathematics Institute, University of Warwick, Coventry CV4 7AL, U.K.
E-mail address: G.Kenison@warwick.ac.uk
Mathematics Institute, University of Warwick, Coventry CV4 7AL, U.K.
E-mail address: R.J.Sharp@warwick.ac.uk