Limiting modular symbol for the shift space

and metric

d(((x_k, e_k))_k,((x⁰_k, e⁰_k)_k)) =

∞

X

i−=1

1 2ⁱ

1−δ_(xi,ei),(x⁰

i,e⁰_i)

. The isomorphism ΣG →ΣG is given by

(e(±[n₁, n₂, ...]), e)7→((∓n₁, e),(∓n₂, τ∓n₁(e),(∓n₃, τ∓n₂(τ∓n₁(e))), ...).

Formulating the limiting modular symbols in terms of the shift space rather than directly in terms of points in R is useful because the shift space (Σ_G, σ) is known to be finitely irreducible (Prop 3.1 [KS07b]). This means that there is a finite set W ⊂Σ^∗_G, where Σ^∗_G is the set of finite admissible words in the alphabet Z^××EG, such that for any a, b∈Z^××E_G there exists w∈W such thatawb∈Σ^∗_G.

It is also shown in [KS07b] that, as elements in ((xi, ei))i ∈ ΣG satisfy ek+1 = τ_xk(e_k), there is a relation in terms of the continued fraction expansion of x =

−sign(x₁)[|x₁|,|x₂|, ...] = [ ˜x₁,x˜₂, ...]

e_k+1≡_Ge1ST^x˜1...ST^x˜k =e1g_k(x) where

g_k(x) = −sign(x₁)pk−1(|x|) (−1)^kp_k(|x|) qk−1(|x|) (−1)^k+1sign(x₁)q_k(|x|)

!

. (4.9)

Similar to equation 4.8, we have the relation describing the action ofg_k(x)on higher- weight modular symbols

g_k(x)|{0, i∞, n, m}_G

=− (

−sign(x₁)pk−1(|x|)

qk−1(|x|),−sign(x₁)p_k(|x|)

qk(|x|), 0 1

−1 −|x_k|

!

. . . 0 1

−1 −|x₁|

! n m

!)

G

. (4.10)

x

γ(t) =x+iexp(−t))

e1(x) i

e1(γ(t2)) e1(γ(t1)) e₁(γ(t))

e1(i∞)

γ(t2) γ(t₁)

Figure 4.2: Definition of the limiting modular symbol for the shift space

It is known that this limit can be equivalently written by approximating the point e₁(x) by its continued fraction expansion (Proposition 4.2 [KS07b])

˜lG(((xk, ek))k) = lim

n→∞

1 2 logqn(|x|)

n

X

i=1

{e_k(i∞), e_k(0)}_G. (4.12) We generalize this picture to the higher-weight setting by putting

˜l_G,n,m: Σ_G→H₁(X_G,Π,(R₁Φ∗Q)^w) ((x_k, e_k))_k7→ lim

t→∞

1

t{i, e₁(x+iexp(−t)), n, m}_G. (4.13) We proceed by obtaining a similar result to equation 4.12, but now with modular symbols of higher weight. Importantly, the result holds everywhere on ΣG.

Theorem 4.3.1. For ((x_k, e_k))_k∈Σ_G we have

˜l_G,N,M(((x_k, e_k))_k) = lim

n→∞

1 2 logq_n(|x|)

∞

X

k=1

{e_k(i∞), e_k(0),˜g_k−1⁻¹ (x)·(N, M)}_G

where ˜gk−1(x) =e₁gk−1(x)e⁻¹_k and we set x=−sign(x₁)[|x₁|,|x₂|, ...].

Proof. The proof follows the strategy outlined in [KS07b], but here we track the additional(n, m)-coordinate data of the higher-weight modular symbol. The general strategy is as follows. We begin by showing that

LG,N,M(((x_k, e_k))_k) := lim

n→∞

1 2 logq_n(|x|)

∞

X

k=1

{e_k(i∞), e_k(0),˜g⁻¹_k−1(N, M)}_G. (4.14) exists if and only if there is a sequence(tn)n∈N tending to infinity such that

n→∞lim 1

t_n{i, e₁(x+ie^−tn), N, M}_G (4.15) exists, and that if either limit exists they coincide. Then, we will show that the limit 4.15 does not depend on the particular sequence (tn)n∈N chosen.

The main idea is to write the geodesic passing throughe1(i∞) and e1(x), in terms of geodesics related to the continued fraction approximants ofx. Define a sequence of points inP¹(Q) by

ξ₁=e₁(i∞) ξ_n=e₁

−sign(x₁)pn−2(|x|) qn−2(|x|)

n≥2 (4.16)

and letωnbe the oriented geodesic inH∪P¹(Q)which starts atξnand ends atξn+1. Next, letl(x)be the oriented vertical geodesic running fromi∞toxand lete1(l(x)) be its image. The image e1(l(x)) is a geodesic starting at ξ1 and ending at e1(x).

Define a sequence of points alonge₁(l(x))by

y_n=ω_n∪e₁(l(x)). (4.17)

ξ₁ =y₁ ξ₃ e₁(x) ξ₄ ξ₂

y₄ y₃ y2

e1(l(x)) ω₁

ω2

ω3

Figure 4.3: Approximatinge₁(l(x))by continued fractions

Note that the oriented geodesic path fromyn to yn+1 is homologous to the geodesic path running fromy_n to ξ_n+1 to y_n+1. Therefore, we have that

{y_n, y_n+1, N, M}_G ={y_n, ξ_n+1, N, M}_G+{ξ_n+1, y_n+1, N, M}_G for alln∈N.

Recall that we have

e_n+1 ≡_G e₁g_n(x)

where g_n for n ≥ 2 is defined in 4.9 and g₁ = id. Therefore, there exists some

˜

g_n(x)∈G such that

˜

gn(x)en+1 =e1g_n(x).

By property 4.2 of the modular symbols, and by directly acting g_n on the points 0 and i∞ by fractional linear transformations, we get

{e_n(i∞), e_n(0),g˜_n−1⁻¹ (x)·(N, M)}_G ={˜gn−1(x)en(i∞),˜gn−1(x)en(0), N, M}_G

={e₁g_n−1(x)(i∞), e₁˜gn−1(x)(0), N, M}_G

={ξ_n, ξ_n+1, N, M}_G.

Using homologous paths and additivity of the modular symbols we find {i, y_n+1, N, M}_G={i, y₂, N, M}_G+{y₂, yn+1, N, M}_G

={i, y₂, N, M}_G+

n

X

k=2

{y_k, y_k+1, N, M}_G

={i, y₂, N, M}_G+

n

X

k=2

({y_k, ξ_k+1, N, M}_G+{ξ_k+1, y_k+1, N, M}_G)

={i, y₂, N, M}_G− {ξ₂, y2, N, M}_G− {y_n+1, ξn+1, N, M}_G+

n+1

X

k=2

{ξ_k, ξk+1, N, M}_G

={i, ξ₁, N, M}_G− {y_n+1, ξ_n+1, N, M}_G+

n+1

X

k=1

{ξ_k, ξ_k+1, N, M}_G

={i, ξ₁, N, M}_G− {y_n+1, ξn+1, N, M}_G+

n+1

X

k=1

{e_k(i∞), e_k(0),g˜⁻¹_k−1·(N, M)}_G Let the sequence tnbe defined by the equation

e1(x+ie^−tn) :=yn.

An argument involving hyperbolic geometry gives an estimate e^−tn ∼(q_n(|x|))², for sufficiently largen(see Lemma 3.3 of [KS07a]).

With this we can complete the first part of the proof, concluding the equivalence of the limits:

L_G,N,M(((x_k, e_k))_k) = lim

n→∞

1 2 logq_n(|x|)

∞

X

k=1

{e_k(i∞), e_k(0),g˜_k−1⁻¹ ·(N, M)}_G

= lim

n→∞

1

t_n({i, y_n+1, N, M}_G+{y_n+1, ξ_n+1, N, M}_G− {i, ξ₁, N, M}_G)

= lim

n→∞

1 tn

({i, y_n, N, M}_G+{y_n, ξn+1, N, M}_G− {i, ξ₁, N, M}_G)

= lim

n→∞

1

t_n{i, y_n, N, M}_G

= lim

n→∞

1

t_n{i, e₁(x+ie^−tn), N, M}_G The second step of the proof is to show that the limitlimn→∞ 1

tn{i, e₁(x+ie^−tn), N, M}_G is independent of the choice of sequence(tn)tending to infinity. Recall that we have

a non-degenerate pairing between the higher-weight modular symbols and the space of cusp forms Sw+2(G)⊕Sw+2(G). Suppose that LG,N,M(((xk, ek))k) exists. For Φ = (Φ₁,Φ₂)∈S_w+2(G)⊕S_w+2(G) arbitrary and t >0, let

α_Φ=hL_G,N,M(((x_k, e_k))_k),Φi and let

nt= sup{n∈N: 2 logqn(|x|)≤t}.

Our aim is to show that for allΦ∈S_w+2(G)⊕S_w+2(G), lim sup

t→∞

h{i, e₁(x+ie^−t), N, M}_G,Φi

t −hP_nt

k=1{e_k(i∞), e_k(0),g˜_k⁻¹(N, M)}_G,Φi 2 logqnt(|x|)

= 0,

which will allow us to conclude that˜l_G,N,M(((x_k, e_k))_k)exists and is equal toL_G,N,M(((x_k, e_k))_k). We obtain a bound following exactly the same strategy as [KS07b], but we repeat it

here for completeness.

lim sup

t→∞

h{i, e₁(x+ie^−t), N, M}_G,Φi

t −

*_nt X

k=1

{e_k(i∞), e_k(0),g˜_k⁻¹(N, M)}_G,Φ +

2 logq_nt(|x|)

= lim sup

t→∞

2 logqnt(|x|)h{i, e₁(x+ie^−t), N, M}_G,Φi 2tlogq_nt(|x|) −

t

* _n

t

X

k=1

{e_k(i∞), e_k(0),˜g⁻¹_k (N, M)}_G,Φ +

2tlogq_nt(|x|)

≤lim sup

t→∞

1

th{i, e₁(x+ie^−t), N, M}_G−

nt

X

k=1

{e_k(i∞), e_k(0),g˜_k⁻¹(N, M)}_G,Φi

+ lim sup

t→∞

2 logqnt(|x|)−t t

* _n

t

X

k=1

{e_k(i∞), e_k(0),g˜_k⁻¹(N, M)}_G,Φ +

2 logqnt(|x|)

≤lim sup

t→∞

const.

t + lim sup

n→∞

log|x_n+1| logq_n(|x|)|α_Φ|

=|α_Φ|lim sup

n→∞

log|x_n+1| logqn(|x|)

where in the last bound we are using the recursion relationq_n(|x|) =|x_n+1|q_n−1(|x|)+

qn−2(|x|).

In the case that α_Φ = 0 for all Φ ∈ S_w+2(G)⊕S_w+2(G), the result follows im- mediately. We will assume wlog that there is someΦsuch thatαΦ >0. In this case,

we will show thatlim sup_{n→∞ log}^log_q^|xn+1|

n(|x|) = 0.To do this, we first observe that

α_Φ = lim

n→∞

_n+1 P

k=1

{e_k(i∞), e_k(0),g˜⁻¹_k (N, M)}_G,Φ

2 logqn+1(|x|)

= lim

n→∞

_n P

k=1

{e_k(i∞), e_k(0),g˜_k⁻¹(N, M)}_G+{e_n+1(i∞), e_n+1(0),g˜⁻¹_n+1(N, M)}_G,Φ

2 logqn(|x|) + 2 log|x_n+1|

= lim

n→∞

_n P

k=1

{e_k(i∞, e_k(0),˜g⁻¹_k (N, M)}_G,Φ



1 +^{h{en+1(i∞),en+1(0),˜g}

−1

n+1(N,M)}G,Φi _n

P

k=1

{e_k(i∞,e_k(0),˜g_k⁻¹(N,M)}_G,Φ





2 logqn(|x|)

1 +_log^log_q^|xn+1|

n(|x|)

=α_Φ lim

n→∞

1 +^{h{en+1(i∞),en+1(0),˜g}

−1

n+1(N,M)}_G,Φi _n

P

k=1

{e_k(i∞),e_k(0),˜g⁻¹_k (N,M)}_G,Φ

1 +_log^log_q^|xn+1|

n(|x|)

.

Suppose for contradiction that lim sup_n→∞ ^log_logq^|xn+1|

n(|x|) > 0. Then there is a subse- quence (n_k)_k such that limk→∞

log|x_nk+1|

logq_nk(|x|) > 0, and hence limk→∞|x_nk+1| = ∞. Since we have assumed thatαΦ>0, we get

1 = lim

k→∞

1 +^{h{enk+1(i∞),enk+1(0),˜g}

−1

nk+1(N,M)}G,Φi

*nk

P

j=1

{e_j(i∞),e_j(0),˜g_j⁻¹(N,M)}_G,Φ +

1 +_log^log|x_q^nk+1|

nk(|x|)

= lim

k→∞

logq_nk(|x|)

*nk

P

j=1

{e_j(i∞), e_j(0),˜g⁻¹_j (N, M)}_G,Φ +

h{e_nk+1(i∞), e_nk+1(0),˜g_n⁻¹

k+1(N, M)}_G,Φi log|x_nk+1|

= 1 αΦ

(0) = 0

This is a contradiction, so we conclude that lim sup_{n→∞ log}^log_q^|xn+1|

n(|x|) = 0.

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