In summary, any element of $\mathfrak{G}_n$ admits in its Mod($\Sigma$)-orbit an element of the form $(0, 0, \dots, 0, 0, 0, \beta)$, and admits also all the elements of the form $(0, 0, \dots, 0, 0, 0, \beta + 2k)$ where $k$ is any integer.

It follows that when $n$ is odd, in which case 2 is a generator of $\mathbb{Z}/n\mathbb{Z}$, that every Mod($\Sigma$)-orbit in $\mathfrak{G}_n$ contains $(0, 0, \dots, 0, 0, 0, 0)$. In other words, there is only one Mod($\Sigma$)-orbit; hence only one Mod$^{\pm}$(Σ)-orbit. The theorem follows in this case from Proposition 3.17.

Let us now assume that $n$ is even. The same argument as above shows that there are at most two Mod($\Sigma$)-orbits in $\mathfrak{G}_n$: the orbit of $(0, 0, \dots, 0, 0, 0, 0)$ and the orbit of $(0, 0, \dots, 0, 0, 0, 1)$. The proof of the Theorem will be finished if we prove that these two orbits are disjoint.

Since $n$ is even, every $\alpha_i$ and $\beta_i$, which is an integer modulo $n$, defines an element $\bar{\alpha}_i$ and $\bar{\beta}_i$ of $\mathbb{Z}/2\mathbb{Z}$. In other words, there is a well-defined morphism $\mathfrak{G}_n \to \mathfrak{G}_2$. This morphism is Mod$^{\pm}$(Σ)-equivariant – this follows from Proposition 4.4 which shows that the action in $\mathfrak{G}_2$ is obtained from taking the action on $\mathfrak{G}_n$ and projecting it to modulo 2. Therefore, if the Mod$^{\pm}$(Σ)-orbits of $(0, 0, \dots, 0, 0, 0, 0)$ and of $(0, 0, \dots, 0, 0, 0, 1)$ are different in $\mathfrak{G}_2$, they are also different in $\mathfrak{G}_n$.

Hence, in order to achieve the proof of the Theorem, we just have to consider the case $n = 2$.

For every element of $\mathfrak{G}_2 = V_1 \oplus \dots \oplus V_g$, let us call vanishing number the number of indices $i$ for which the components in $V_i$ are zero. More precisely, the vanishing number is this integer modulo 2. For example, the vanishing number for $(0, 0, \dots, 0, 0, 0, 0)$ is the class modulo 2 of 0, whereas the vanishing number for $(0, 0, \dots, 0, 0, 0, 1)$ is the class modulo 2 of 1: they are different.

**Claim: The vanishing number is constant along Mod($\Sigma$)-orbits.**

Let us prove the claim: clearly, the vanishing number does not change under the action of $A_i$ or $B_i$, since they act in $V_i$ as elements of SL(2, $\mathbb{Z}/2\mathbb{Z}$).

Let us consider $C_i$ ($1 \le i \le g - 1$). It acts trivially on every $V_j$ except maybe $V_i$ and $V_{i+1}$. We refer to the explicit formula for the action of $C_i$ on $(\alpha_i, \beta_i, \alpha_{i+1}, \beta_{i+1})$ given in the beginning of the proof of this theorem.

- If $\alpha_i = \alpha_{i+1} = 1$, the components in $V_i$ and $V_{i+1}$ are nonzero, and the same is true after applying $C_i$ since we still have after the action $\alpha_i = \alpha_{i+1} = 1$: the vanishing number remains the same.

- If $\alpha_i \neq \alpha_{i+1}$, then, since we are in $\mathbb{Z}/2\mathbb{Z}$, we have $\alpha_{i+1} - \alpha_i + 1 = 0$, and it follows that $C_i$ acts trivially on such an element: the vanishing number remains the same in this case too.

- If $\alpha_i = \alpha_{i+1} = 0$: then we have:

$$C_i(0, \beta_i, 0, \beta_{i+1}) = (0, \beta_i + 1, 0, \beta_{i+1} + 1)$$

Hence, if $\beta_i = \beta_{i+1} = 0$, or if $\beta_i = \beta_{i+1} = 1$, the vanishing number before and after $C_i$ differ by 2, hence is the same modulo 2. If $\beta_i \neq \beta_{i+1}$, one component vanishes and not the other, and the same is true after applying $C_i$.

We have proved the Claim. Since the vanishing numbers of $(0, 0, \dots, 0, 0, 0, 0)$ and $(0, 0, \dots, 0, 0, 0, 1)$ are different, they cannot be in the same Mod($\Sigma$)-orbit.

The only remaining step is to show that the vanishing number is also preserved by orientation reversing elements of Mod$^{\pm}$(Σ). For this, we just have to prove that it is true for one of them. Let us consider the horizontal plane $P$ such that the surface is symmetric relatively to the symmetry $s$ in $P$. We can isotop every $B_i$ such that they are all contained in $P$, hence preserved by $s$ (including their orientation), and such that every $A_i$ is orthogonal to $P$ and preserved by $s$, but with the reversed orientation. We refer to figure 2: we can choose $s$ so that it preserves the loops $A_i, B_i$ in figure 2. One can assume, and we do, that the base point $x_0$ is in $P$, hence fixed by $s$, and that for every $i$, the portion $\hat{a}_i$ (respectively $\hat{b}_i$) of the loop $\bar{a}_i$ (respectively $\bar{b}_i$) joining $x_0$ to $A_i$ (respectively $B_i$) lies above $P$, and, in addition, in the case of $\bar{a}_i$, this portion reaches $A_i$ at its intersection with $P$.

Then, the composition of $\hat{a}_i$ (respectively $\hat{b}_i$) with its image under $s$ provides a loop $\hat{a}_i^s$ (respectively $\hat{b}_i^s$) based at $x_0$. It follows from these choices that $\bar{b}_i$ (respectively $\bar{a}_i$) is conjugated by $\hat{b}_i^s$ (respectively $\hat{a}_i^s$) to itself (respectively to its inverse).

Therefore, according to Lemma 4.2, and since $s$ is orientation reversing, the action of $[s]$ on $\mathfrak{G}_2$ (or $\mathfrak{G}_n$) is given by, for all $j$:
In summary, any element of $\mathfrak{G}_n$ admits in its Mod($\Sigma$)-orbit an element of the form $(0, 0, \dots, 0, 0, 0, \beta)$, and admits also all the elements of the form $(0, 0, \dots, 0, 0, 0, \beta + 2k)$ where $k$ is any integer.

It follows that when $n$ is odd, in which case 2 is a generator of $\mathbb{Z}/n\mathbb{Z}$, that every Mod($\Sigma$)-orbit in $\mathfrak{G}_n$ contains $(0, 0, \dots, 0, 0, 0, 0)$. In other words, there is only one Mod($\Sigma$)-orbit; hence only one Mod$^{\pm}$(Σ)-orbit. The theorem follows in this case from Proposition 3.17.

Let us now assume that $n$ is even. The same argument as above shows that there are at most two Mod($\Sigma$)-orbits in $\mathfrak{G}_n$: the orbit of $(0, 0, \dots, 0, 0, 0, 0)$ and the orbit of $(0, 0, \dots, 0, 0, 0, 1)$. The proof of the Theorem will be finished if we prove that these two orbits are disjoint.

Since $n$ is even, every $\alpha_i$ and $\beta_i$, which is an integer modulo $n$, defines an element $\bar{\alpha}_i$ and $\bar{\beta}_i$ of $\mathbb{Z}/2\mathbb{Z}$. In other words, there is a well-defined morphism $\mathfrak{G}_n \to \mathfrak{G}_2$. This morphism is Mod$^{\pm}$(Σ)-equivariant – this follows from Proposition 4.4 which shows that the action in $\mathfrak{G}_2$ is obtained from taking the action on $\mathfrak{G}_n$ and projecting it to modulo 2. Therefore, if the Mod$^{\pm}$(Σ)-orbits of $(0, 0, \dots, 0, 0, 0, 0)$ and of $(0, 0, \dots, 0, 0, 0, 1)$ are different in $\mathfrak{G}_2$, they are also different in $\mathfrak{G}_n$.

Hence, in order to achieve the proof of the Theorem, we just have to consider the case $n = 2$.

For every element of $\mathfrak{G}_2 = V_1 \oplus \dots \oplus V_g$, let us call vanishing number the number of indices $i$ for which the components in $V_i$ are zero. More precisely, the vanishing number is this integer modulo 2. For example, the vanishing number for $(0, 0, \dots, 0, 0, 0, 0)$ is the class modulo 2 of 0, whereas the vanishing number for $(0, 0, \dots, 0, 0, 0, 1)$ is the class modulo 2 of 1: they are different.

**Claim: The vanishing number is constant along Mod($\Sigma$)-orbits.**

Let us prove the claim: clearly, the vanishing number does not change under the action of $A_i$ or $B_i$, since they act in $V_i$ as elements of SL(2, $\mathbb{Z}/2\mathbb{Z}$).

Let us consider $C_i$ ($1 \le i \le g - 1$). It acts trivially on every $V_j$ except maybe $V_i$ and $V_{i+1}$. We refer to the explicit formula for the action of $C_i$ on $(\alpha_i, \beta_i, \alpha_{i+1}, \beta_{i+1})$ given in the beginning of the proof of this theorem.

- If $\alpha_i = \alpha_{i+1} = 1$, the components in $V_i$ and $V_{i+1}$ are nonzero, and the same is true after applying $C_i$ since we still have after the action $\alpha_i = \alpha_{i+1} = 1$: the vanishing number remains the same.

- If $\alpha_i \neq \alpha_{i+1}$, then, since we are in $\mathbb{Z}/2\mathbb{Z}$, we have $\alpha_{i+1} - \alpha_i + 1 = 0$, and it follows that $C_i$ acts trivially on such an element: the vanishing number remains the same in this case too.

- If $\alpha_i = \alpha_{i+1} = 0$: then we have:

$$C_i(0, \beta_i, 0, \beta_{i+1}) = (0, \beta_i + 1, 0, \beta_{i+1} + 1)$$

Hence, if $\beta_i = \beta_{i+1} = 0$, or if $\beta_i = \beta_{i+1} = 1$, the vanishing number before and after $C_i$ differ by 2, hence is the same modulo 2. If $\beta_i \neq \beta_{i+1}$, one component vanishes and not the other, and the same is true after applying $C_i$.

We have proved the Claim. Since the vanishing numbers of $(0, 0, \dots, 0, 0, 0, 0)$ and $(0, 0, \dots, 0, 0, 0, 1)$ are different, they cannot be in the same Mod($\Sigma$)-orbit.

The only remaining step is to show that the vanishing number is also preserved by orientation reversing elements of Mod$^{\pm}$(Σ). For this, we just have to prove that it is true for one of them. Let us consider the horizontal plane $P$ such that the surface is symmetric relatively to the symmetry $s$ in $P$. We can isotop every $B_i$ such that they are all contained in $P$, hence preserved by $s$ (including their orientation), and such that every $A_i$ is orthogonal to $P$ and preserved by $s$, but with the reversed orientation. We refer to figure 2: we can choose $s$ so that it preserves the loops $A_i, B_i$ in figure 2. One can assume, and we do, that the base point $x_0$ is in $P$, hence fixed by $s$, and that for every $i$, the portion $\hat{a}_i$ (respectively $\hat{b}_i$) of the loop $\bar{a}_i$ (respectively $\bar{b}_i$) joining $x_0$ to $A_i$ (respectively $B_i$) lies above $P$, and, in addition, in the case of $\bar{a}_i$, this portion reaches $A_i$ at its intersection with $P$.

Then, the composition of $\hat{a}_i$ (respectively $\hat{b}_i$) with its image under $s$ provides a loop $\hat{a}_i^s$ (respectively $\hat{b}_i^s$) based at $x_0$. It follows from these choices that $\bar{b}_i$ (respectively $\bar{a}_i$) is conjugated by $\hat{b}_i^s$ (respectively $\hat{a}_i^s$) to itself (respectively to its inverse).

Therefore, according to Lemma 4.2, and since $s$ is orientation reversing, the action of $[s]$ on $\mathfrak{G}_2$ (or $\mathfrak{G}_n$) is given by, for all $j$:
<figure><img src="image_8.png" /><figcaption>Figure 6: $g-1$ non separating curves on the surface.</figcaption></figure>

$$[s] * \nu(a_j) = \nu(a_j)$$

$$[s] * \nu(\bar{b}_j) = -\nu(\bar{b}_j)$$

This obviously preserves the vanishing number. Thus concludes the proof of Theorem 4.5. $\square$

We end this Section with a proposition showing that the image in $\mathrm{Aff}(\mathfrak{G}_n)$ of $\mathrm{Mod}(\Sigma)$ is isomorphic to $\mathrm{Sp}(2g, \mathbb{Z}/n\mathbb{Z})$.

**Proposition 4.7.** *The Torelli group acts trivially on $\mathfrak{G}_n$.*

*Proof.* By definition, an element of the Torelli group is an element of $\mathrm{Mod}^{\pm}(\Sigma)$ which acts trivially in homology. In particular it is orientation preserving. It also follows that its linear part is the identity. But it could still act on $\mathfrak{G}_n$ as a translation. However, by a Theorem of Johnson ([Joh]) combined with a Theorem of Powell in the case $g=2$ ([Pow]), the Torelli group is generated by:

- Dehn twists along separating non-homotopically trivial simple closed curves in the case $g=2$,

- compositions $T_1T_2^{-1}$ where $T_1$ and $T_2$ are left Dehn twists along simple closed curves that are boundary of a compact surface of genus 1 embedded in $\Sigma$ (in the case $g \ge 3$).

Consider the red simple closed curves $D_1, \dots, D_{g-1}$ depicted in Figure 6. In the case $g=2$, there is only one such a curve $D_1$, and every separating, homotopically non-trivial simple closed curve in $\Sigma$ is the image of $D_1$ under some homeomorphism. Therefore, it follows that in this case the Torelli group is generated by the conjugates of $T_1$.

In the case $g \ge 3$ any pair of simple closed curves bounding a genus 1 surface embedded in $\Sigma$ is the image under some homeomorphism of the region of $\Sigma$ between two successive loops $D_i$ and $D_{i+1}$. Therefore, the Torelli group is generated by conjugates of one of the $T_iT_{i+1}^{-1}$.

Therefore, in order to prove the proposition we just have to show that every $T_i$ acts trivially on $\mathfrak{G}_n$.

But every $D_i$ are disjoint from the loops $A_i$ and $B_i$. Therefore the Dehn twists along $D_i$ preserves the conjugacy classes of $\bar{a}_i$ and $\bar{b}_i$. It follows that every $T_i$ acts trivially on $\mathfrak{G}_n$. $\square$

**Remark 4.8.** According to proposition 4.7, the action of $\mathrm{Mod}(\Sigma)$ on $\mathfrak{G}_n$ induces a (faithful) affine action of $\mathrm{Sp}(2g, \mathbb{Z}/n\mathbb{Z})$ on the torus $\mathfrak{G}_n \approx (\mathbb{Z}/n\mathbb{Z})^{2g}$ whose linear part is the canonical representation of $\mathrm{Sp}(2g, \mathbb{Z}/n\mathbb{Z})$. But this action is affine and admits a non-trivial translation part $[\tau]$, that represents a non-trivial element of $H^1(\mathrm{Sp}(2g, \mathbb{Z}/n\mathbb{Z}), (\mathbb{Z}/n\mathbb{Z})^{2g})$.

This feature is in contrast with the fact that $H^1(\mathrm{Sp}(2g, \mathbb{Z}), (\mathbb{Z})^{2g})$ vanishes.

**Remark 4.9.** In [Gi], E. Giroux considered a related problem: the classification of contact structures on the circle bundle $M$ up to isotopy and up to conjugation by a diffeomorphism. A case of interest for us is the case of universally tight contact structures, with wrapping number $-n$. E. Giroux shows that they are all isotopic to the pull-back $q^*\alpha_0$ by some finite covering $q: M \to M_1(\Sigma)$ along the fibers, where $\alpha_0$ is the contact 1-form on $M_1(\Sigma)$ whose Reeb flow is the geodesic flow (for some hyperbolic Riemannian metric). According to Ghys' Theorem (see Theorem 3.3), Anosov flows on $M$ are always isotopic to the Reeb flow of such a structure. Clearly, an isotopy between contact structures provides an isotopy between their Reeb flows, i.e. Anosov flows, and vice-versa. Similarly, there is a one-to-one correspondence between conjugacy classes of these contact structures and orbital equivalence classes of Anosov foliations.

As a matter of fact, Giroux obtains a classification of isotopy classes similar as we do here: the action of $K_1$ on these contact structures is simply transitive. Actually, it follows from Giroux's work and the results in the present paper that these two classification problems are equivalent.
<figure><img src="image_8.png" /><figcaption>Figure 6: $g-1$ non separating curves on the surface.</figcaption></figure>

$$[s] * \nu(a_j) = \nu(a_j)$$

$$[s] * \nu(\bar{b}_j) = -\nu(\bar{b}_j)$$

This obviously preserves the vanishing number. Thus concludes the proof of Theorem 4.5. $\square$

We end this Section with a proposition showing that the image in $\text{Aff}(\mathfrak{G}_n)$ of $\text{Mod}(\Sigma)$ is isomorphic to $\text{Sp}(2g, \mathbb{Z}/n\mathbb{Z})$.

**Proposition 4.7.** *The Torelli group acts trivially on $\mathfrak{G}_n$.*

*Proof.* By definition, an element of the Torelli group is an element of $\text{Mod}^{\pm}(\Sigma)$ which acts trivially in homology. In particular it is orientation preserving. It also follows that its linear part is the identity. But it could still act on $\mathfrak{G}_n$ as a translation. However, by a Theorem of Johnson ([Joh]) combined with a Theorem of Powell in the case $g=2$ ([Pow]), the Torelli group is generated by:

- Dehn twists along separating non-homotopically trivial simple closed curves in the case $g=2$,

- compositions $T_1T_2^{-1}$ where $T_1$ and $T_2$ are left Dehn twists along simple closed curves that are boundary of a compact surface of genus 1 embedded in $\Sigma$ (in the case $g \ge 3$).

Consider the red simple closed curves $D_1, \dots, D_{g-1}$ depicted in Figure 6. In the case $g=2$, there is only one such a curve $D_1$, and every separating, homotopically non-trivial simple closed curve in $\Sigma$ is the image of $D_1$ under some homeomorphism. Therefore, it follows that in this case the Torelli group is generated by the conjugates of $T_1$.

In the case $g \ge 3$ any pair of simple closed curves bounding a genus 1 surface embedded in $\Sigma$ is the image under some homeomorphism of the region of $\Sigma$ between two successive loops $D_i$ and $D_{i+1}$. Therefore, the Torelli group is generated by conjugates of one of the $T_iT_{i+1}^{-1}$.

Therefore, in order to prove the proposition we just have to show that every $T_i$ acts trivially on $\mathfrak{G}_n$.

But every $D_i$ are disjoint from the loops $A_i$ and $B_i$. Therefore the Dehn twists along $D_i$ preserves the conjugacy classes of $\bar{a}_i$ and $\bar{b}_i$. It follows that every $T_i$ acts trivially on $\mathfrak{G}_n$. $\square$

**Remark 4.8.** According to proposition 4.7, the action of $\text{Mod}(\Sigma)$ on $\mathfrak{G}_n$ induces a (faithful) affine action of $\text{Sp}(2g, \mathbb{Z}/n\mathbb{Z})$ on the torus $\mathfrak{G}_n \approx (\mathbb{Z}/n\mathbb{Z})^{2g}$ whose linear part is the canonical representation of $\text{Sp}(2g, \mathbb{Z}/n\mathbb{Z})$. But this action is affine and admits a non-trivial translation part $[\tau]$, that represents a non-trivial element of $H^1(\text{Sp}(2g, \mathbb{Z}/n\mathbb{Z}), (\mathbb{Z}/n\mathbb{Z})^{2g})$.

This feature is in contrast with the fact that $H^1(\text{Sp}(2g, \mathbb{Z}), (\mathbb{Z})^{2g})$ vanishes.

**Remark 4.9.** In [Gi], E. Giroux considered a related problem: the classification of contact structures on the circle bundle $M$ up to isotopy and up to conjugation by a diffeomorphism. A case of interest for us is the case of universally tight contact structures, with wrapping number $-n$. E. Giroux shows that they are all isotopic to the pull-back $q^*\alpha_0$ by some finite covering $q: M \to M_1(\Sigma)$ along the fibers, where $\alpha_0$ is the contact 1-form on $M_1(\Sigma)$ whose Reeb flow is the geodesic flow (for some hyperbolic Riemannian metric). According to Ghys' Theorem (see Theorem 3.3), Anosov flows on $M$ are always isotopic to the Reeb flow of such a structure. Clearly, an isotopy between contact structures provides an isotopy between their Reeb flows, i.e. Anosov flows, and vice-versa. Similarly, there is a one-to-one correspondence between conjugacy classes of these contact structures and orbital equivalence classes of Anosov foliations.

As a matter of fact, Giroux obtains a classification of isotopy classes similar as we do here: the action of $K_1$ on these contact structures is simply transitive. Actually, it follows from Giroux's work and the results in the present paper that these two classification problems are equivalent.
Giroux furthermore claims to have classified conjugacy classes, and to show that the number of con-
jugacy classes of these contact structures is the number $\varphi(n)$ of divisors of $n$ ([Gi, Theorem 3.1 and
Proposition 3.10]). Hence, this statement is in contradiction with our Theorem 4.5.

The point is that Giroux’s proof is not completely correct at its concluding step: in the proof of Proposition 3.10, he implicitly assumes that the action of Mod($\Sigma$) on the space $\mathcal{O}_n$ of index $n$ subgroups along the fiber is linear, whereas this action is genuinely affine, as shown in the present article.

References

[An] D. V. Anosov, *Geodesic flows on closed Riemannian manifolds with negative curvature*, Proc. Steklov Inst. Math. **90** (1969).

[Ba1] T. Barbot, *Caractérisation des flots d'Anosov en dimension 3 par leurs feuilletages faibles*, Ergod. Th. Dynam. Sys. **15** (1995) 247-270.

[Ba2] T. Barbot, *Flots d'Anosov sur les variétés graphées au sens de Waldhausen*, Ann. Inst. Fourier Grenoble **46** (1996) 1451-1517.

[Ba3] T. Barbot, *Plane affine geometry of Anosov flows*, Ann. Sci. Ecole Norm. Sup. **34** (2001) 871-889.

[Ba-Ma] T. Barthelme, K. Mann, *Orbit equivalences of $\mathbb{R}$-covered Anosov flows and hyperbolic-like actions on the line*, to appear in Geom. Topol., arxiv: 2012.11811.

[BFeM] T. Barthelme, S. Fenley, K. Mann, *Anosov flows with the same periodic orbits*, submitted, arxiv: 2308.02098.

[BFrM] T. Barthelme, S. Frankel, K. Mann, *Orbit equivalence of pseudo-Anosov flows*, submitted, arxiv: 2211.10505.

[Ep] Epstein, D. *Periodic flows on three-dimensional manifold*, Ann. of Math. **95** (1972) 66-82.

[Fa-Ma] B. Farb, D. Margalit, *A primer on mapping class groups*, Princeton Mathematical Series **49** (2012).

[Fen] S. Fenley, *Anosov flows in 3-manifolds*, Ann. of Math. **139** (1994) 79-115.

[Fe-Mo] S. Fenley and L. Mosher, *Quasigeodesic flows in hyperbolic 3-manifolds*, Topology **40** (2001) 503-537.

[Gh1] E. Ghys, *Flots d'Anosov sur les 3-variétés fibrées en cercles*, Ergod. Th and Dynam. Sys. **4** (1984) 67-80.

[Gh2] E. Ghys, *Groupes d'homéomorphismes du cercle et cohomologie bornée*, Contemporary Mathematics **58** (1987) 81-106.

[Gh3] E. Ghys, *Groups acting on the circle*, Enseign. Math. (2) **47** (2001) 329-407.

[Gi] E. Giroux, *Structures de contact sur les variétés fibrées en cercles au-dessus d'une surface*, Comm. Math. Helv. **76** (2001) 218-262.

[Gro] M. Gromov, *Three remarks on geodesic dynamics and fundamental group*, Enseign. Math. (1) **46** (2000) 391-402

[Ha-Ka] B. Hasselblatt and A. Katok, *Introduction to the modern theory of duynamical systems*, Cambridge Univ. Press, 1995.

[Hu-Ka] S. Hurder, A. Katok, *Differentiability, rigidity and Godbillon-Vey classes for Anosov flows*, Inst. Hautes Études Sci. Publ. Math. **72** (1990), 5-61.

[Joh] D.L. Johnson, *Homeomorphisms of a surface which act trivially on homology*, Proc. Amer. Soc., **75** (1979) 119-125.

[Lic] W. B. R. Lickorish, *A finite set of generators for the homeotopy group of a 2-manifold*, Proc. Cambridge Philos. Soc., **60** (1964) 769-778.

[Pow] J. Powell, *Two theorems on the mapping class group of a surface*, Proc. Amer. Soc., **68** (1978) 347-350.

[Rand] O. Randal-Williams, *Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces*, J. Topol., **7** (2014), no. 1, 155-186.

[Sou] J. Souto, *A remark on the action of the mapping class group on the unit tangent bundle*, Ann. Fac. Sci. Toulouse Math. **19** (2010) 589-601.

[Wald] F. Waldhausen, *On irreducible 3-manifolds which are sufficiently large*, Ann. of Math. **87** (1968) 56-88.

THIERRY BARBOT, AVIGNON UNIVERSITÉ, LMA, CAMPUS JEAN-HENRI FABRE, 301, RUE BARUCH DE SPINOZA, F-84
916 AVIGNON CEDEX 9.

Email address: thierry.barbot@univ-avignon.fr

SÉRGIO FENLEY, FLORIDA STATE UNIVERSITY, TALLAHASSEE, FL 32306-4510, USA

Email address: fenley@math.fsu.edu
Giroux furthermore claims to have classified conjugacy classes, and to show that the number of con-
jugacy classes of these contact structures is the number $\varphi(n)$ of divisors of $n$ ([Gi, Theorem 3.1 and
Proposition 3.10]). Hence, this statement is in contradiction with our Theorem 4.5.

The point is that Giroux’s proof is not completely correct at its concluding step: in the proof of Proposition 3.10, he implicitly assumes that the action of Mod($\Sigma$) on the space $\mathcal{O}_n$ of index $n$ subgroups along the fiber is linear, whereas this action is genuinely affine, as shown in the present article.

References

[An] D. V. Anosov, *Geodesic flows on closed Riemannian manifolds with negative curvature*, Proc. Steklov Inst. Math. **90** (1969).

[Ba1] T. Barbot, *Caractérisation des flots d'Anosov en dimension 3 par leurs feuilletages faibles*, Ergod. Th. Dynam. Sys. **15** (1995) 247-270.

[Ba2] T. Barbot, *Flots d'Anosov sur les variétés graphées au sens de Waldhausen*, Ann. Inst. Fourier Grenoble **46** (1996) 1451-1517.

[Ba3] T. Barbot, *Plane affine geometry of Anosov flows*, Ann. Sci. Ecole Norm. Sup. **34** (2001) 871-889.

[Ba-Ma] T. Barthelme, K. Mann, *Orbit equivalences of $\mathbb{R}$-covered Anosov flows and hyperbolic-like actions on the line*, to appear in Geom. Topol., arxiv: 2012.11811.

[BFeM] T. Barthelme, S. Fenley, K. Mann, *Anosov flows with the same periodic orbits*, submitted, arxiv: 2308.02098.

[BFrM] T. Barthelme, S. Frankel, K. Mann, *Orbit equivalence of pseudo-Anosov flows*, submitted, arxiv: 2211.10505.

[Ep] Epstein, D. *Periodic flows on three-dimensional manifold*, Ann. of Math. **95** (1972) 66-82.

[Fa-Ma] B. Farb, D. Margalit, *A primer on mapping class groups*, Princeton Mathematical Series **49** (2012).

[Fen] S. Fenley, *Anosov flows in 3-manifolds*, Ann. of Math. **139** (1994) 79-115.

[Fe-Mo] S. Fenley and L. Mosher, *Quasigeodesic flows in hyperbolic 3-manifolds*, Topology **40** (2001) 503-537.

[Gh1] E. Ghys, *Flots d'Anosov sur les 3-variétés fibrées en cercles*, Ergod. Th and Dynam. Sys. **4** (1984) 67-80.

[Gh2] E. Ghys, *Groupes d'homéomorphismes du cercle et cohomologie bornée*, Contemporary Mathematics **58** (1987) 81-106.

[Gh3] E. Ghys, *Groups acting on the circle*, Enseign. Math. (2) **47** (2001) 329-407.

[Gi] E. Giroux, *Structures de contact sur les variétés fibrées en cercles au-dessus d'une surface*, Comm. Math. Helv. **76** (2001) 218-262.

[Gro] M. Gromov, *Three remarks on geodesic dynamics and fundamental group*, Enseign. Math. (1) **46** (2000) 391-402

[Ha-Ka] B. Hasselblatt and A. Katok, *Introduction to the modern theory of duynamical systems*, Cambridge Univ. Press, 1995.

[Hu-Ka] S. Hurder, A. Katok, *Differentiability, rigidity and Godbillon-Vey classes for Anosov flows*, Inst. Hautes Études Sci. Publ. Math. **72** (1990), 5-61.

[Joh] D.L. Johnson, *Homeomorphisms of a surface which act trivially on homology*, Proc. Amer. Soc., **75** (1979) 119-125.

[Lic] W. B. R. Lickorish, *A finite set of generators for the homeotopy group of a 2-manifold*, Proc. Cambridge Philos. Soc., **60** (1964) 769-778.

[Pow] J. Powell, *Two theorems on the mapping class group of a surface*, Proc. Amer. Soc., **68** (1978) 347-350.

[Rand] O. Randal-Williams, *Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces*, J. Topol., **7** (2014), no. 1, 155-186.

[Sou] J. Souto, *A remark on the action of the mapping class group on the unit tangent bundle*, Ann. Fac. Sci. Toulouse Math. **19** (2010) 589-601.

[Wald] F. Waldhausen, *On irreducible 3-manifolds which are sufficiently large*, Ann. of Math. **87** (1968) 56-88.

THIERRY BARBOT, AVIGNON UNIVERSITÉ, LMA, CAMPUS JEAN-HENRI FABRE, 301, RUE BARUCH DE SPINOZA, F-84 916 AVIGNON CEDEX 9.

Email address: thierry.barbot@univ-avignon.fr

SÉRGIO FENLEY, FLORIDA STATE UNIVERSITY, TALLAHASSEE, FL 32306-4510, USA

Email address: fenley@math.fsu.edu