Posted by: matheuscmss | February 24, 2012


Today we will make some preparations towards the application of Avila-Viana simplicity criterion to the case of the Kontsevich-Zorich cocycle over the {SL(2,\mathbb{R})}-orbits of square-tiled surfaces (along the lines of a forthcoming article by C.M., M. Möller, and J.-C. Yoccoz). Of course, given that the simplicity criterion was stated in terms of locally constant cocycles over complete shifts on finitely or countably many symbols, we need to discuss how to “reduce”, or more precisely, to code, the Kontsevich-Zorich cocycle over the {SL(2,\mathbb{R})}-orbits of square-tiled surfaces to the setting of complete shifts. In fact, the main purpose of this post (corresponding to J.-C. Yoccoz 7th lecture) is the presentation of an adequate coding of the Teichmüller flow and Kontsevich-Zorich cocycle over the {SL(2,\mathbb{R})}-orbits of square-tiled surfaces.

Let {\pi:M\rightarrow\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2} be a reduced origami (i.e., an origami whose periods generate the lattice {\mathbb{Z}\oplus i\mathbb{Z}}). Let’s consider the Kontsevich-Zorich cocycle {G_{KZ}^t} over the Teichmüller flow restricted to the unit tangent bundle of the Teichmüller surface (“curve”) associated to {M}.

More concretely, we consider (the unit tangent bundle) {SL(2,\mathbb{R})/SL(M)}, where {SL(M)} is the Veech group of {M} (that is, the stabilizer of {M} under the action of {SL(2,\mathbb{R})} on the moduli space of Abelian differentials). Recall that {SL(M)} is a finite-index subgroup of {SL(2,\mathbb{Z})} (as {M} is a reduced origami). In this language, the Teichmüller geodesic flow is the action of

\displaystyle g_t=\left(\begin{array}{cc}e^t & 0 \\ 0 & e^{-t} \end{array}\right)

We begin our discussion with the case of {SL(2,\mathbb{R})/SL(2,\mathbb{Z})}. In the sequel, we will think of {SL(2,\mathbb{R})/SL(2,\mathbb{Z})} as the space of normalized (i.e., unit covolume) lattices of {\mathbb{R}^2}, and we will select an appropriate fundamental domain. Here, it is worth to point out that we’re not going to consider the lift to {SL(2,\mathbb{R})} of the “classical” fundamental domain {\mathcal{F}=\{z\in\mathbb{H}: |z|\geq 1, |\textrm{Re}z|\leq 1/2\}} of the action of {SL(2,\mathbb{Z})} on the hyperbolic plane {\mathbb{H}}. Indeed, as we will see below, our choice of fundamental domain is not {SO(2,\mathbb{R})}-invariant, while any fundamental domain obtained by lifting to {SL(2,\mathbb{R})} a fundamental domain of {\mathbb{H}/SL(2,\mathbb{Z})} must be {SO(2,\mathbb{R})}-invariant (as {\mathbb{H}/SL(2,\mathbb{Z})=SO(2,\mathbb{R})\backslash SL(2,\mathbb{R})/SL(2,\mathbb{Z})}).

Definition 1 A lattice {L\subset\mathbb{R}^2} is irrational if {L} intersect the coordinate axis {x} and {y} precisely at the origin {0\in\mathbb{R}^2}.

Equivalently, {L} is irrational if and only if the orbit {g_t(L)} doesn’t diverge (neither in the past nor in the future) to the cusp of {SL(2,\mathbb{R})/SL(2,\mathbb{Z})}.

Our choice of fundamental domain will be guided by the following fact:

Proposition 2 Let {L} be a normalized irrational lattice. Then, there exists an unique basis {\{v_1=(\lambda_1,\tau_1), v_2=(\lambda_2,\tau_2)\}} of {L} such that exactly one of the two possibilities below occur:

  • “Top” case: {\lambda_2\geq 1>\lambda_1>0} and {0<\tau_2<-\tau_1};
  • “Bottom” case: {\lambda_1\geq 1>\lambda_2>0} and {0<-\tau_1<\tau_2}.

Proof: Consider the following open unit area squares of the plane: {Q^+:=(0,1)\times (0,1)} and {Q^-=(0,1)\times (-1,0)}.

Observe that {Q^{\pm}} can’t contain two linearly independent vectors of {L}: indeed, if {v_1, v_2\in Q^{\pm}\cap L}, and {v_1, v_2} are linearly independent, then {0<|\det(v_1,v_2)|<1}, a contradiction with the fact that {\det(v_1,v_2)\in\mathbb{Z}} (as {L} is normalized).

On the other hand, we claim that {Q^+\cup Q^-} contains (at least) one vector of {L}. In fact, since {L} is normalized and irrational, one would have that {L} is disjoint from some convex symmetric open set (strictly) containing the closure of {Q^+\cup Q^-\cup (-Q^+)\cup(-Q^-)}, a contradiction with Minkowski theorem (that any convex symmetric set {C\subset \mathbb{R}^d} of volume {\textrm{vol}(C)>2^d} intersects any normalized lattice of {\mathbb{R}^d}).

In particular, we have three possibilities:

  • (a) {Q^+\cap L\neq\emptyset}, {Q^-\cap L=\emptyset};
  • (b) {Q^+\cap L=\emptyset}, {Q^-\cap L\neq\emptyset};
  • (c) {Q^+\cap L\neq\emptyset}, {Q^-\cap L\neq\emptyset}.

Because the first two cases are similar, we’ll treat only items (b) and (c).

We start by item (b), that is, {Q^-\cap L\neq\emptyset} but {Q^+\cap L=\emptyset}. In this situation, we select a primitive {v_1=(\lambda_1,\tau_1)\in Q^-\cap L}, so that

\displaystyle 0<\lambda_1<1 \quad \textrm{ and } 0<-\tau_1<1

Next, we select {v_2=(\lambda_2,\tau_2)} such that

  • {\{v_1, v_2\}} is a direct basis, i.e., {\det(v_1,v_2)=\lambda_1\tau_2-\lambda_2\tau_1=1};
  • {\tau_2>0} is minimal.

Then, {\tau_1+\tau_2<0}: otherwise we could replace {v_2} by {v_2+v_1} to contradict the minimality. Thus, {\lambda_2>0} (as {\{v_1, v_2\}} is a direct basis, and {0<\tau_2<-\tau_1<1} forces {0<\lambda_1\tau_2<1}). Since {Q^+\cap L=\emptyset}, we have that {\lambda_2\geq 1}, and hence {\{v_1,v_2\}} is a basis of {L} fitting the requirements of the top case. Now, we verify the uniqueness of such {\{v_1, v_2\}}. Firstly, if {\{v_1'=(\lambda_1',\tau_1'), v_2'=(\lambda_2',\tau_2')\}} fits the requirements {\lambda_2'\geq 1>\lambda_1'>0} and {0<-\tau_1'<\tau_2'} of the bottom case, then the relation

\displaystyle 1=\lambda_1'\tau_2'-\lambda_2'\tau_1'

implies that {\tau_2'<1}, and, a fortiori, {v_2'\in Q^+\cap L}, a contradiction with our assumptions in item (b). Secondly, if {\{v_1', v_2'\}} fits the requirements {\lambda_2'\geq 1>\lambda_1'>0} and {0<\tau_2'<-\tau_1'} of the top case, then {v_1'\in Q^-\cap L}, and, therefore, {v_1'=v_1} (as {Q^-} can’t contain two linearly independent vectors of {L}). Now, we write {v_2'=v_2+n v_1}, and we notice that, since {0<\tau_2'<-\tau_1'=-\tau_1}, {0<\tau_2<-\tau_1}, and {\tau_2'=\tau_2+n\tau_1}, one has {n=0}, i.e., {v_2'=v_2}, and the analysis of item (b) is complete.

It remains only to analyze item (c). Take {V_1=(\Lambda_1, T_1)\in Q^-\cap L} and {V_2=(\Lambda_2, T_2)\in Q^+\cap L} primitive vectors. We have that

\displaystyle 2>\det(V_1,V_2)=\Lambda_1 T_2 - \Lambda_2 T_1 >0

and, since {\det(V_1,V_2)\in\mathbb{Z}}, it follows that {\Lambda_1 T_2 - \Lambda_2 T_1 = 1}. Furthermore, {T_1+T_2\neq 0} because {L} is irrational. Assume {T_1+T_2<0} (the other case {T_1+T_2>0} is analogous). Then, we set {v_2:= V_2} and {v_1:=V_1+nV_2} where {n\geq 1} is the largest integer such that {T_1+nT_2<0}. We have that {v_1=(\lambda_1,\tau_1)}, {v_2=(\lambda_2,\tau_2)} verifies {0<\lambda_2,\tau_2<1} (as {v_2:=V_2\in Q^+}), and {0<-\tau_1<\tau_2} (as {n} was taken to be the largest possible). Furthermore, {\lambda_1\geq 1}, as, otherwise, {v_1=V_1+nV_2} (recall that {n\geq 1}) and {V_1} would be linearly independent vectors of {L} inside {Q^-}, a contradiction. In resume, {\{v_1,v_2\}} is a basis of {L} meeting the requirements of the bottom case. Now, we check the uniqueness of such {\{v_1', v_2'\}}. Here, since the argument is the same one used for item (b), we will illustrate only the bottom case {\lambda_1'\geq 1>\lambda_2'>0}, {0<-\tau_1'<\tau_2'}. In this situation, {v_2'\in Q^-} (as {L} is normalized), so that {v_2'=v_2}. Then, we write {v_1'=v_1+nv_2}, and we conclude that {n=0} because {0<-\tau_1'<\tau_2'=\tau_2}, {0<-\tau_1<\tau_2}, and {\tau_1'=\tau_1+n\tau_2}. \Box

Using this proposition, we can describe the Teichmüller geodesic flow {g_t=\left(\begin{array}{cc}e^t & 0 \\ 0 & e^{-t} \end{array}\right)} on the space {SL(2,\mathbb{R})/SL(2,\mathbb{Z})} of normalized lattices as follows. Let {L_0} be a normalized irrational lattice, and let {(v_1,v_2)} be the basis of {L_0} given by the proposition above, i.e., the top, resp. bottom, condition. Then, we see that the basis {(g_t v_1, g_t v_2)} of {L_t:=g_t L_0} satisfies the top, resp. bottom condition for all {t <t^*}, where {\lambda_1 e^{t^*}=1} in the top case, resp. {\lambda_2 e^{t^*}=1} in the bottom case.

However, at time {t^*}, the basis {\{v_1^*=g_{t^*} v_1, v_2^*=g_{t^*} v_2\}} of {L_0} ceases to fit the requirements of the proposition above, but we can remedy this problem by changing the basis: for instance, if the basis {\{v_1,v_2\}} of the initial lattice {L_0} has top type, then it is not hard to check that

\displaystyle v_1'=v_1^* \quad \textrm{ and } \quad v_2'=v_2^*-a v_1^*

where {a=\lfloor\lambda_2/\lambda_1\rfloor} is a basis of {L_{t^*}} of bottom type.

Here, we observe that the quantity {\alpha:=\lambda_1/\lambda_2\in (0,1)} giving the ratios of the first coordinates of the vectors {g_t v_1, g_t v_2} forming a top type basis of {L_t} for any {0\leq t<t^*} is related to the integer a by the formula

\displaystyle a=\lfloor 1/\alpha\rfloor

Also, the new quantity {\alpha'} giving the ratio of the first coordinates of the vectors {v_1', v_2'} forming a bottom type basis of {L_{t^*}} is related to {\alpha} by the formula

\displaystyle \alpha'=\lambda_2'/\lambda_1' = \{1/\alpha\}:=G(\alpha)

where {G} is the so-called Gauss map. In this way, we find the classical relationship between the geodesic flow on the modular surface {SL(2,\mathbb{R})/SL(2,\mathbb{Z})} and the continued fraction algorithm.

At this stage, we’re ready to code the Teichmüller flow over the unit tangent bundle of the Teichmüller surface {SL(2,\mathbb{R})/SL(M)} associated to a reduced origami.

-Coding the geodesic flow on {SL(2,\mathbb{R})/SL(M)}

Let {\Gamma(M)} be the following graph: the set of its vertices is

\displaystyle \textrm{Vert}(\Gamma(M)) = \{SL(2,\mathbb{Z})-\textrm{orbit of } M \}\times \{t,b\}

\displaystyle = \{M=M_1,\dots, M_r\}\times \{t,b\}

and its arrows are

\displaystyle (M_i,c)\stackrel{\gamma_{a,i,c}}{\rightarrow} (M_j,\overline{c})

where {a\in\mathbb{N}}, {a\geq 1}, {c\in\{t,b\}}, {\overline{c}= b} (resp. {t}) if {c=t} (resp. {b}), and

\displaystyle M_j=\left\{\begin{array}{cl}\left(\begin{array}{cc} 1 & a \\ 0 & 1\end{array}\right) M_i, & \textrm{if } c=t \\ \left(\begin{array}{cc} 1 & 0 \\ a & 1\end{array}\right) M_i, & \textrm{if } c=b \end{array}\right.

Notice that this graph has finitely many vertices but countably many arrows. Using this graph, we can code irrational orbits of the flow {g_t} on {SL(2,\mathbb{R})/SL(M)} as follows. Given {m_0\in SL(2,\mathbb{R})}, let {L_{st}=\mathbb{Z}^2} be the standard lattice and put {m_0 L_{st} = L_0}. Also, let us denote {m_t=g_t m_0}.

By Proposition 2, there exists an unique {h_0\in SL(2,\mathbb{Z})} such that {v_1=m_0 h_0^{-1}(e_1)}, {v_2 = m_0 h_0^{-1}(e_2)} satisfying the conditions of the proposition (here, {\{e_1, e_2\}} is the canonical basis of {\mathbb{R}^2}). Denote by {c} the type (top or bottom) of the basis {\{v_1,v_2\}} of {L_0}. We assign to {m_0} the vertex {(M_i:=h_0 M, c)\in\textrm{Vert}(\Gamma(M))}.

For sake of concreteness, let’s assume that {c=t} (top case). Recalling the notations introduced after the proof of Proposition 2, we notice that the lattice {L_{t^*}} associated to {m_{t^*}} has a basis of bottom type

\displaystyle v_1' = g_{t^*} m_0 h_0^{-1}(e_1) = g_{t^*} m_0 h_1^{-1}(e_1)


\displaystyle v_2' = g_{t^*} m_0 h_0^{-1}(e_2-a e_1) = g_{t^*} m_0 h_1^{-1}(e_2)

where {h_1=h_* h_0} and

\displaystyle h_* = \left(\begin{array}{cc}1 & a \\ 0 & 1\end{array}\right)

In other words, starting from the vertex {(M_i,t)} associated to the initial point {m_0}, after running the geodesic flow for a time {t^*}, we end up with the vertex {(M_j,b)} where {M_j= h_* M_i}. Equivalently, the piece of trajectory from {m_0} to {g_{t^*} m_0} is coded by the arrow

\displaystyle (M_i,t)\stackrel{\gamma_{i,a,t}}{\rightarrow}(M_j,b)

Evidently, we can iterate this procedure (by replacing {L_0} by {L_{t^*}}) in order to code the entire orbit {g_t m_0} by a succession of arrows. However, this coding has the “inconvenient” (with respect to the setting of Avila-Viana simplicity criterion) that it is not associated to a complete shift but only a subshift (as we do not have the right to concatenate two arrows {\gamma} and {\gamma'} unless the endpoint of {\gamma} coincides with the start of {\gamma'}).

Fortunately, this little difficulty is easy to overcome: in order to get a coding by a complete shift, it suffices to fix a vertex {p^*\in\textrm{Vert}(\Gamma(M))} and consider exclusively concatenations of loops based at {p^*}. Of course, we pay a price here: since there may be some orbits of {g_t} whose coding is not a concatenation of loops based on {p^*}, we’re throwing away some orbits in this new way of coding. But, it is not hard to see that the (unique, Haar) {SL(2,\mathbb{R})}-invariant probability {\mu} on {SL(2,\mathbb{R})/SL(M)} gives zero weight to the orbits that we’re throwing away, so that this new coding still captures most orbits of {g_t} (from the point of view of {\mu}). In any case, this allows to code {g_t} by a complete shift whose (countable) alphabet is constituted of (minimal) loops based at {p^*}.

Once we know how to code our flow {g_t} by a complete shift, the next natural step (in view of Avila-Viana criterion) is the verification of the bounded distortion condition of the invariant measure induced by {\mu} on the complete shift. This is the content of the next section.

-Verification of the bounded distortion condition-

As we saw above, the coding of the geodesic flow (and modulo the stable manifolds, that is, the “{\tau}-coordinates” [vertical coordinates]) is the dynamical system

\displaystyle \textrm{Vert}(\Gamma(M))\times ((0,1)\cap(\mathbb{R}-\mathbb{Q}))\rightarrow\textrm{Vert}(\Gamma(M))\times ((0,1)\cap(\mathbb{R}-\mathbb{Q}))

given by {(p,\alpha)\mapsto (p',G(\alpha))} where {G(\alpha)=\{1/\alpha\}=\alpha'} is the Gauss map and {p\stackrel{\gamma_{a,p}}{\rightarrow}p'} with {a=\lfloor1/\alpha\rfloor}. In this language, {\mu} becomes (up to normalization) the Gauss measure {dt/(1+t)} on each copy {\{p\}\times (0,1)}, {p\in \textrm{Vert}(\Gamma(M))}, of the unit interval {(0,1)}.

Now, for sake of concreteness, let us fix {p^*} a vertex of top type. Given {\gamma} a loop based on {p^*}, i.e., a word on the letters of the alphabet of the coding leading to a complete shift, we denote by {I(\gamma)\subset (0,1)} the interval corresponding to {\gamma}, that is, the interval {I(\gamma)} consisting of {\alpha\in (0,1)} such that the concatenation of loops (based at {p^*}) coding the orbit of {(p^*,\alpha)} starts by the word {\gamma}.

In this setting, the measure induced by {\mu} on the complete shift is easy to express: by definition, the measure of the cylinder {\Sigma(\gamma)} corresponding to concatenations of loops (based at {p^*}) starting by {\gamma} is the Gauss measure of the interval {I(\gamma)} up to normalization. Because the Gauss measure is equivalent to the Lebesgue measure (as its density {1/(1+t)} satisfies {1/2\leq1/(1+t)\leq 1} in {(0,1)}), we conclude that the measure of {\Sigma(\gamma)} is equal to

\displaystyle |I(\gamma)|:=\textrm{Lebesgue measure of } I(\gamma)

up to a multiplicative constant.

In particular, it follows that the bounded distortion condition for the measure induced by {\mu} on the complete shift is equivalent to the existence of a constant {C>0} such that

\displaystyle C^{-1}|I(\gamma_0)|\cdot|I(\gamma_1)|\leq |I(\gamma)|\leq C|I(\gamma_0)|\cdot |I(\gamma_1)| \ \ \ \ \ (1)

for every {\gamma=\gamma_0\gamma_1}.

In resume, this reduces the bounded distortion condition to the problem of understanding the interval {I(\gamma)}. Here, by the usual properties of the continued fraction, it is not hard to show that {I(\gamma)} is a Farey interval

\displaystyle I(\gamma)=\left(\frac{p}{q}, \frac{p+p'}{q+q'}\right)


\displaystyle \left(\begin{array}{cc}p' & p \\ q' & q\end{array}\right)\in SL(2,\mathbb{Z})

being t-reduced, i.e., {0<p'\leq p,q'\leq q}.

Consequently, from this description, we recover the classical fact that

\displaystyle \frac{1}{2q^2}\leq |I(\gamma)|=\frac{1}{q(q+q')}\leq \frac{1}{q^2} \ \ \ \ \ (2)

Given {\gamma=\gamma_0\gamma_1}, and denoting by {\left(\begin{array}{cc}p_0' & p_0 \\ q_0' & q_0\end{array}\right)}, resp. {\left(\begin{array}{cc}p_1' & p_1 \\ q_1' & q_1\end{array}\right)}, resp. {\left(\begin{array}{cc}p' & p \\ q' & q\end{array}\right)} the matrices associated to {\gamma_0}, resp. {\gamma_1}, resp. {\gamma}, it is not hard to check that

\displaystyle \left(\begin{array}{cc}p' & p \\ q' & q\end{array}\right)=\left(\begin{array}{cc}p_0' & p_0 \\ q_0' & q_0\end{array}\right)\left(\begin{array}{cc}p_1' & p_1 \\ q_1' & q_1\end{array}\right)

so that {q=q_0'p_1+q_0q_1}. Because these matrices are t-reduced, we have that

\displaystyle q_0q_1\leq q\leq 2q_0q_1

Therefore, in view of (1) and (2), the bounded distortion condition follows.

Once we know that the basis dynamics (Teichmüller geodesic flow on {SL(2,\mathbb{R})/SL(M)}) is coded by a complete shift equipped with a probability measure with bounded distortion, we can pass to the study of the Kontsevich-Zorich cocycle in terms of the coding.

-Cocycle over the complete shift induced by {G_{KZ}^t}

Let {(M_i,[\textrm{t, resp. b}])\stackrel{\gamma_{a,i,t}}{\rightarrow}(M_j,[\textrm{b, resp. t}])} be an arrow of {\Gamma(M)} and denote by {A:M_i\rightarrow M_j} an affine map of derivative {\left(\begin{array}{cc}1 & a \\ 0 & 1\end{array}\right)}, resp. {\left(\begin{array}{cc}1 & 0 \\ a & 1\end{array}\right)}. Of course, {A} is only well-defined up to automorphisms of {M_i} and/or {M_j}. In terms of translation structures, given {g\in SL(2,\mathbb{R})} and a translation structure {\zeta} on {M}, the identity map {\textrm{id}:(M,\zeta)\rightarrow (M,g\zeta)} is an affine map of derivative {g}.

Given {\gamma} a path in {\Gamma(M)} obtained by concatenation {\gamma=\gamma_1\dots\gamma_{\ell}}, and starting at {(M_i,c)} and ending at {(M_j,c')}, one has, by functoriality, {A_{\gamma}:M_i\rightarrow M_j} an affine map given by {A_{\gamma}=A_{\gamma_{\ell}}\dots A_{\gamma_1}}.

Suppose now that {\gamma} is a loop based at {(M,c)}. Then, by definition, the derivative {A_{\gamma}\in SL(M)}. For our subsequent discussions, an important question is: what matrices of {SL(M)} can be obtained in this way?

In this direction, we recall the following definition (already encountered in the previous section):

Definition 3 We say that {A=\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)\in SL(2,\mathbb{Z})} is

  • t-reduced if {0< a\leq b,c < d};
  • b-reduced if {0<d\leq b,c <a}.

Observe that the product of two t-reduced (resp. b-reduced) matrices is also t-reduced (resp. b-reduced), i.e., these conditions are stable by products.

The following statement is the answer to the question above:

Corollary 4 The matrices associated to the loops {\gamma} based at the vertex {(M,c)} are precisely the c-reduced matrices of {SL(M)}.

Indeed, this is a corollary to the next proposition:

Proposition 5 A matrix {A} is t-reduced if and only if there exists {k\geq 1} and {a_1,\dots, a_{2k}\geq 1} such that

\displaystyle A=\left(\begin{array}{cc}1 & 0 \\ a_{2k} & 1\end{array}\right)\left(\begin{array}{cc}1 & a_{2k-1} \\ 0 & 1\end{array}\right)\dots \left(\begin{array}{cc}1 & 0 \\ a_2 & 1\end{array}\right)\left(\begin{array}{cc}1& a_1 \\ 0 & 1\end{array}\right)

Furthermore, the decomposition above is unique.

Of course, one has similar statements for b-reduced matrices (by conjugation by the matrix {\left(\begin{array}{cc}0& 1 \\ 1 & 0\end{array}\right)\in GL(2,\mathbb{Z})}).

Actually, this proposition follows from the following slightly more general fact:

Proposition 6 Let {A=\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)\in SL(2,\mathbb{Z})} with {a,b,c,d\geq 0}. Then, there exists an unique decomposition

\displaystyle A=\left(\begin{array}{cc}1 & 0 \\ a_{2k} & 1\end{array}\right)\left(\begin{array}{cc}1 & a_{2k-1} \\ 0 & 1\end{array}\right)\dots \left(\begin{array}{cc}1 & 0 \\ a_2 & 1\end{array}\right)\left(\begin{array}{cc}1& a_1 \\ 0 & 1\end{array}\right)

with {a_i\geq 0} for all {i}, and {a_i>0} if {1<i<2k}

Assuming the validity of Proposition 6, we can derive Proposition 5 as follows. As one can easily check, it suffices to rule out the possibilities {a_{2k}=0} or {a_1=0} in the decomposition above. We treat only the case {a_{2k}=0} as {a_1=0} is analogous. If {a_{2k}=0}, we would have

\displaystyle A=\left(\begin{array}{cc}1 & a_{2k-1} \\ 0 & 1\end{array}\right)\left(\begin{array}{cc}a' & b' \\ c' & d'\end{array}\right) = \left(\begin{array}{cc}a'+a_{2k-1}c' & b'+a_{2k-1}d' \\ c' & d'\end{array}\right)

with {d'<b'+a_{2k-1}d'} (as {a_{2k-1}\geq 1}), that is, {A} is not t-reduced.

Concerning the proof of Proposition 6, while it is not difficult (essentially an “Euclidean division algorithm”-like argument), we prefer to omit it in order to present the following related proposition:

Proposition 7 {A\in SL(2,\mathbb{Z})} is conjugated (in {SL(2,\mathbb{Z})}) to a t-reduced matrix if and only if its trace {\textrm{tr}(A)>2}.

We close this post with the proof of this proposition.

Proof: Since

\displaystyle \left(\begin{array}{cc}1 & 0 \\ a_2 & 1\end{array}\right)\left(\begin{array}{cc}1& a_1 \\ 0 & 1\end{array}\right)=\left(\begin{array}{cc}1 & a_1 \\ a_2 & 1+a_1a_2\end{array}\right)

has trace {2+a_1a_2>2} whenever {a_1,a_2\geq 1}, we have that if {A} is conjugated to a t-reduced matrix, then, by Proposition 5, {\textrm{tr}(A)>2}.

Conversely, given {A\in SL(2,\mathbb{Z})} with {\textrm{tr}(A)>2}, its eigenvalues satisfy {\lambda>1>\lambda^{-1}}. Let {\{e_u,e_s\}\subset\mathbb{R}^2} be a direct normalized basis with {A(e_u)=\lambda e_u}, {A(e_s)=\lambda^{-1}e_s}.

There exists {g\in SL(2,\mathbb{Z})} such that {g e_1=\alpha e_u+\beta e_s}, {g e_2=\gamma e_u + \delta e_s} with {\alpha,\gamma,\delta>0}, {\beta<0}. Geometrically, these conditions correspond to the following picture:

In this situation, the matrix {A':=g^{-1} A g} has nonnegative coefficients. By Proposition 6, we have the following possibilities:

  • (a) {a_1,a_{2k}>0}
  • (b) {a_1=a_{2k}=0}
  • (c) {a_1>0}, {a_{2k}=0}
  • (d) {a_1=0}, {a_{2k}>0}

Evidently, the proof is complete in the cases (a) and (b). Also, the cases (c) and (d) are similar, so that the argument is finished once we treat (c): in this situation, we observe that

\displaystyle \left(\begin{array}{cc}1 & a_{2k-1} \\ 0 & 1\end{array}\right)\dots \left(\begin{array}{cc}1 & 0 \\ a_2 & 1\end{array}\right)\left(\begin{array}{cc}1& a_1 \\ 0 & 1\end{array}\right)

is conjugated to

\displaystyle \left(\begin{array}{cc}1 & a_{2k-1} \\ 0 & 1\end{array}\right)^{-1}\cdot\left(\begin{array}{cc}1 & a_{2k-1} \\ 0 & 1\end{array}\right)\dots \left(\begin{array}{cc}1 & 0 \\ a_2 & 1\end{array}\right)\left(\begin{array}{cc}1& a_1 \\ 0 & 1\end{array}\right)\cdot\left(\begin{array}{cc}1 & a_{2k-1} \\ 0 & 1\end{array}\right)

that is,

\displaystyle \left(\begin{array}{cc}1 & 0 \\ a_{2k-2} & 1\end{array}\right)\dots \left(\begin{array}{cc}1 & 0 \\ a_2 & 1\end{array}\right)\left(\begin{array}{cc}1& a_1+a_{2k-1} \\ 0 & 1\end{array}\right)

a t-reduced matrix (by Proposition 5). \Box

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