Free ${\Bbb Z}^p$-actions on the three dimensional torus

TL;DR

This paper demonstrates that for each natural number p ≥ 2, the Lefschetz fixed point theorem's application to free ${Z}^p$-actions on the 3-torus is optimal, and constructs such actions with prescribed homological properties.

Contribution

It establishes the existence of free ${Z}^p$-actions on ${T}^3$ with specified homological actions, showing the optimality of Lefschetz fixed point theorem in this context.

Findings

Existence of free ${Z}^p$-actions with prescribed homological action

Optimality of Lefschetz fixed point theorem for these actions

Normal form characterization of the actions

Abstract

We show that for each natural $p\geq 2$, the Lefschetz fixed point theorem is optimal when applied to ${\Bbb Z}^{p}$-actions by homeomorphisms on the three dimensional torus ${\Bbb T}^3$. More precisely, we show that for a spectrally unitary ${\Bbb Z}^p$-action ${\bf A}$ on the first homology group $H_1({\Bbb T}^3,{\Bbb Z})$ with trivial fixed point set, there exists a free ${\Bbb Z}^p$-action by real analytic diffeomorphisms of ${\Bbb T}^3$ whose induced ${\Bbb Z}^p$-action on $H_1({\Bbb T}^3,{\Bbb Z})$ is the action ${\bf A}$. In particular, we establish the normal form for this type of actions.

Equations

$$\mathrm{Fix}(\mathbf{A})=\{k\in {\mathbb{Z}}^q|\mathbf{A}(\ell )k=k\text{ for all }\ell \in {\mathbb{Z}}^p\}.$$

$$A(w_{1})=\left[\begin{array}[]{ccc}1&a&b\\ 0&-1&0\\ 0&0&-1\end{array}\right],\;A(w_{2})=\left[\begin{array}[]{ccc}-1&0&c\\ 0&-1&d\\ 0&0&1\end{array}\right]$$

$$A(e_{1})=\left[\begin{array}[]{ccc}1&a&b\\ 0&-1&0\\ 0&0&-1\end{array}\right]\mbox{ and }A(e_{2})=\left[\begin{array}[]{ccc}-1&0&c\\ 0&-1&d\\ 0&0&1\end{array}\right]$$

$$N=\left[\begin{array}[]{ccc}1&a&b\\ 0&-1&0\\ 0&0&-1\end{array}\right]\mbox{, }M=\left[\begin{array}[]{ccc}-1&0&c\\ 0&-1&d\\ 0&0&1\end{array}\right]$$

$$A(e_{1})=\left[\begin{array}[]{cccc}1&a&0&0\\ 0&-1&0&0\\ 0&0&0&-1\\ 0&0&1&-2\end{array}\right],A(e_{2})=\left[\begin{array}[]{cccc}-1&0&0&0\\ 0&-1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right]$$

$${\mathbb{C}}^3=\mathrm{Fix}(N)\oplus \mathrm{Fix}(M)\oplus \mathrm{Fix}(NM).$$

$$N=\left[\begin{array}[]{ccc}1&\tilde{a}_{12}&\tilde{a}_{13}\\ 0&-1&\tilde{a}_{23}\\ 0&0&-1\end{array}\right]$$

$$M=\left[\begin{array}[]{ccc}-1&\tilde{b}_{12}&\tilde{a}_{13}\\ 0&\tilde{b}_{22}&\tilde{b}_{23}\\ 0&\tilde{b}_{32}&\tilde{b}_{33}\end{array}\right].$$

$$U^{-1}BU=\left[\begin{array}[]{cc}-1&b_{23}\\ 0&b_{33}\end{array}\right]$$

$$\left[\begin{array}[]{ccc}1&0&0\\ 0&U&\end{array}\right]^{-1}N\left[\begin{array}[]{ccc}1&0&0\\ 0&U&\end{array}\right]=\left[\begin{array}[]{ccc}1&a_{12}&a_{13}\\ 0&-1&0\\ 0&0&-1\end{array}\right]$$

$$\left[\begin{array}[]{ccc}1&0&0\\ 0&U&\end{array}\right]^{-1}M\left[\begin{array}[]{ccc}1&0&0\\ 0&U&\end{array}\right]=\left[\begin{array}[]{ccc}-1&b_{12}&b_{13}\\ 0&-1&b_{23}\\ 0&0&b_{33}\end{array}\right]$$

$${\mathbb{C}}^3=\mathrm{Fix}(N)\oplus \mathrm{Fix}(M)\oplus \mathrm{Fix}(NM)$$

$$v_k=\sum _{i\ne k}{\lambda }_i{v}_i.$$

$${\tilde{\phi }}_1(x,y,z)$$

$${\tilde{\phi }}_2(x,y,z)$$

$${\tilde{\phi }}_j(x,y,z)$$

$${\tilde{\phi }}_i\circ {\tilde{\phi }}_j-{\tilde{\phi }}_j\circ {\tilde{\phi }}_i\in {\mathbb{Z}}^3$$

$${\tilde{\phi }}_2\circ {\tilde{\phi }}_1(x,y,z)$$

$${\tilde{\phi }}_1\circ {\tilde{\phi }}_2(x,y,z)$$

$$f_2\circ R-f_2=a{g}_2+f_1\circ T_{\frac{1}{2}}+f_1\text{mbox}and{g}_2+g_2\circ R=g_1+g_1\circ T_{\frac{1}{2}}.$$

$${\tilde{\phi }}_j\circ {\tilde{\phi }}_1(x,y,z)$$

$${\tilde{\phi }}_1\circ {\tilde{\phi }}_j(x,y,z)$$

$$f_j\circ R-f_j=a{g}_j\text{mbox}and{g}_j\circ R=-g_j.$$

$${\tilde{\phi }}_j\circ {\tilde{\phi }}_2(x,y,z)$$

$${\tilde{\phi }}_2\circ {\tilde{\phi }}_j(x,y,z)$$

$$f_j\circ T_{\frac{1}{2}}=-f_j\text{mbox}and{g}_j\circ T_{\frac{1}{2}}=-g_j.$$

$${\tilde{\phi }}_1^2(x,y,z)=(x+{\alpha }_1\cos 2\pi z-\frac{a}{2}{\alpha }_1\sin 2\pi z+2r,y+{\alpha }_1\sin 2\pi z,z)$$

$${\tilde{\phi }}_2^2(x,y,z)=(x+{\alpha }_2\cos 2\pi z-\frac{a}{2}{\alpha }_2\sin 2\pi z+\frac{c}{2},y+{\alpha }_2\sin 2\pi z+\frac{d}{2},z+1),$$

$${\tilde{\phi }}_1^{2{\ell }_1}\circ {\tilde{\phi }}_2^{2{\ell }_2}\circ ...\circ {\tilde{\phi }}_p^{{\ell }_p}(x,y,z)=$$

$$(x+\left({\sum }_{j=1}^p{\ell }_j{\alpha }_j\right)\cos 2\pi z-\frac{a}{2}\left({\sum }_{j=1}^p{\ell }_j{\alpha }_j\right)\sin 2\pi z+2r+{\ell }_1\frac{c}{2},$$

$$y+\left({\sum }_{j=1}^p{\ell }_j{\alpha }_j\right)\sin 2\pi z+{\ell }_2\frac{d}{2},z+{\ell }_2).$$

Full text

Keywords: Actions of higher rank abelian groups, free actions, finite orbits. AMS classification: 37B05, 57M60, 57R30.

1 Introduction.

Let $p,q$ be natural numbers. By a $\mathbb{Z}^{p}$-action of class $C^{r}$(resp. analytic) on the torus $\mathbb{T}^{q}=\mathbb{R}^{q}/\mathbb{Z}^{q}$ we mean a homomorphism $\varphi$ of $\mathbb{Z}^{p}$ into Diff${}^{r}\left(\mathbb{T}^{q}\right)$, where Diff${}^{r}\left(\mathbb{T}^{q}\right)$ is the group of $C^{r}$ diffeomorphisms of the torus $\mathbb{T}^{q}$, $0\leq r\leq\infty$ (resp. analytic). It is well known that the $\mathbb{Z}^{p}$-actions $\varphi$ of class $C^{r}$ on the torus $\mathbb{T}^{q}$ are closely related to the $\mathbb{Z}^{p}$-actions on $H_{1}(\mathbb{T}^{q},\mathbb{Z})=\mathbb{Z}^{q}$. In fact, any $\mathbb{Z}^{p}$-action of class $C^{r}$ on the torus $\mathbb{T}^{q}$ induces a $\mathbb{Z}^{p}$-action $\varphi_{*}$ on $H_{1}(\mathbb{T}^{q},\mathbb{Z})$, which acts by automorphisms of $\mathbb{Z}^{q}$, called the *induced action * by $\varphi$. When $\varphi$ is free, that is $\varphi(\ell)$ has no fixed points, for each $\ell\in\mathbb{Z}^{p}\smallsetminus\{0\}$, it follows from the Lefschetz fixed point theorem that $1$ is an eigenvalue of $\varphi_{*}(\ell)$ for each $\ell\in\mathbb{Z}^{p}$.

We say that a $\mathbb{Z}^{p}$-action ${\bf A}$ on $H_{1}(\mathbb{T}^{q},\mathbb{Z})$ is spectrally unitary if ${\bf A}$ acts by automorphisms of $\mathbb{Z}^{q}$ and $1$ belongs to the spectrum of ${\bf A}(\ell)$ for each $\ell\in\mathbb{Z}^{p}$. Notice that any free $\mathbb{Z}^{p}$-action $\varphi$ on the torus $\mathbb{T}^{q}$ induces a spectrally unitary $\mathbb{Z}^{p}$-action on $\mathbb{Z}^{q}$.

We will denote by ${\rm Fix}({\bf A})$ the set of fixed points of ${\bf A}$, i.e.,

[TABLE]

In this paper, we are interested in the study of the existence of finite orbits for $\mathbb{Z}^{p}$-actions on the torus $\mathbb{T}^{q}$. This study is related to the classification of the cohomologically rigid $\mathbb{Z}^{p}$-actions on manifolds, in the sense that the cohomology group of the action is isomorphic to de Rham cohomology of the torus $\mathbb{T}^{p}$, see [5]. In fact, for $p=1$, the cohomologically rigid diffeomorphisms of the torus $T^{q}$, where $q\leq 3$, are $C^{\infty}$-conjugate to Diophantine translations  [7]. Recently Forni, Flaminio, and Rodriguez-Hertz  [1] proved that the only cohomologically rigid affine transformations of a homogenous space are conjugated to toral diophantine translations.

For $p\geq 2$, dos Santos in  [6] proved that the cohomologically rigid $\mathbb{Z}^{p}$-actions on $\mathbb{T}^{q}$, $1\leq q\leq 2$ are smooth conjugations of affine cohomologically rigid actions. In order to extend this result to actions on $\mathbb{T}^{3}$, one difficulty is knowing whether the induced homology of these actions has an invariant direction. More precisely, in  [6, Problem 2.8.], dos Santos poses the following stronger question: let $\varphi:\mathbb{Z}^{p}\rightarrow$Diff${}^{r}\left(\mathbb{T}^{q}\right)$, $r\geq 0$, be an action and $\varphi_{*}$ be its induced action on $H_{1}(\mathbb{T}^{q},\mathbb{Z})$ such that ${\rm Fix}({\bf\varphi_{*}})=\{0\}$. Does $\varphi$ have a finite orbit?

When $q=1$ this question has an affirmative answer, this follows from Poincaré classification for homeomorphisms of the circle  [3]. The question also has an affirmative answer for $\mathbb{Z}^{p}$-actions of class $C^{1}$ on the torus $\mathbb{T}^{2}$, which follows from Franks, Handel, and Parwani’s work  [2]. In  [6] the author observed that every affine $\mathbb{Z}^{p}$-action $\varphi$ on the torus $T^{q}$ satisfying $\mbox{\rm Fix}(\varphi_{*})=\{0\}$ has a finite orbit. We recall that a $\mathbb{Z}^{p}$-action $\varphi$ on the torus $\mathbb{T}^{q}$ is affine if $\varphi(\ell)$ is an affine transformation of the torus $\mathbb{T}^{q}$ for each $\ell\in\mathbb{Z}^{p}$.

In this paper, we give a negative answer to the stronger question for $q=3$, which we establish in the following theorem.

Theorem 1

Let $p\geq 2$ and let ${\bf A}$ be a spectrally unitary $\mathbb{Z}^{p}$-action on $H_{1}(\mathbb{T}^{3},\mathbb{Z})$. Then there is a free real analytic $\mathbb{Z}^{p}$-action $\varphi$ on the torus $\mathbb{T}^{3}$ whose induced action on the first homology group of $\mathbb{T}^{3}$ is ${\bf A}$.

Any $\mathbb{Z}^{p}$-action ${\bf A}$ on $\mathbb{Z}^{q}$ that acts by automorphisms of $\mathbb{Z}^{q}$, induce an action $A:\mathbb{Z}^{p}\rightarrow{\rm GL}(q,\mathbb{Z})$. Conversely, any action $A:\mathbb{Z}^{p}\rightarrow{\rm GL}(q,\mathbb{Z})$ defines an action by automorphisms. The image of ${\bf A}$ is by definition the abelian group of automorphisms of $\mathbb{Z}^{q}$ generated by ${\bf A}(\ell)$, where $\ell\in\mathbb{Z}^{p}$ and is isomorphic to the subgroup of matrices $A(\mathbb{Z}^{p})\subset{\rm GL}(q,\mathbb{Z})$.

To prove Theorem 1, we developed the normal form for the spectrally unitary $\mathbb{Z}^{p}$-actions on $H_{1}(\mathbb{T}^{3},\mathbb{Z})$ with trivial fixed point set, which is established in the following theorem.

Theorem 2

Let $p\geq 2$ and let ${\bf A}$ be a spectrally unitary $\mathbb{Z}^{p}$-action on $\mathbb{Z}^{3}$ satisfying ${\rm Fix}({\bf A})=\{0\}$. Then, there exists a basis $\{w_{1},...,w_{p}\}$ of $\mathbb{Z}^{p}$ such that after a conjugation by an automorphisms of $\mathbb{Z}^{3}$, ${\bf A}$ has the following normal form in the canonical basis of $\mathbb{Z}^{3}$

[TABLE]

where $ad+2(b+c)=0$ and $A(\ell)$ is the identity matrix for all $\ell=\sum^{p}_{i=1}\ell_{i}w_{i}\in\mathbb{Z}^{p}$ satisfying $\ell_{1}=\ell_{2}=0$. In particular, the image of ${\bf A}$ is isomorphic to the Klein four-group.

When ${\rm Fix}({\bf A})\neq\{0\}$ Theorem 1 follows from  [8, Theorem 3], whose proof is based on the classification of the unipotent $\mathbb{Z}^{p}$-actions on $\mathbb{Z}^{3}$. In each of these cases we constructed a free affine $\mathbb{Z}^{p}$-action.

For $p\geq 3$ and $q\geq 4$, not every unipotent action admits a free affine action. For $p=2$, we do not know.

The rest of this paper is organized as follows. In section 2, we characterize, under certain hypotheses, the abelian subgroups of ${\rm GL}(q,\mathbb{Z})$ with two generators (Lemma 2.2 and Lemma 2.5). Using these results, we obtain the normal form for spectrally unitary $\mathbb{Z}^{p}$-actions on $H_{1}(\mathbb{T}^{q},\mathbb{Z})$ (Theorem 2). In section 3, we establish the proof of our principal result (Theorem 1). In section 4, we reformulate the stronger question for $\mathbb{Z}^{p}$-actions on the torus $\mathbb{T}^{q}$ preserving the Haar measure.

2 Spectrally unitary $\mathbb{Z}^{p}$- actions on $\mathbb{Z}^{3}$ with trivial fixed point set.

In this section we will prove Theorem 2. For this, we need to establish some results for spectrally unitary $\mathbb{Z}^{p}$-actions on $\mathbb{Z}^{3}$ with trivial fixed point set.

Definition 2.1

Let ${\bf A}$ and ${\bf B}$ be two $\mathbb{Z}^{p}$-actions on $\mathbb{Z}^{q}$ that acts by automorphisms of $\mathbb{Z}^{q}$ . We say that ${\bf A}$ is conjugate to ${\bf B}$ if there is an automorphism $h$ of $\mathbb{Z}^{q}$ such that ${\bf B}(\ell)\circ h=h\circ{\bf A}(\ell)$ for all $\ell\in\mathbb{Z}^{p}$.

Let $A$ and $B$: $\mathbb{Z}^{p}\rightarrow{\rm GL}(q,\mathbb{Z})$ be two actions. We say that $A$ and $B$ are conjugate if there is $P\in{\rm GL}(q,\mathbb{Z})$ such that $B(\ell)P=PA(\ell)$ for all $\ell\in\mathbb{Z}^{p}$. Two actions ${\bf A}$ and ${\bf B}$ are conjugate via automorphism if and only if the corresponding actions $A$ and $B$ are conjugate via an element of ${\rm GL}(q,\mathbb{Z})$.

A fundamental example in the proof of our main result is the $\mathbb{Z}^{2}$-action ${\bf A}$ on $\mathbb{Z}^{3}$ generated, in the canonical basis $\{e_{1},e_{2}\}$ of $\mathbb{Z}^{2}$, by the matrices

[TABLE]

where $ad+2(b+c)=0$. Since ${\bf A}(2e_{1})={\bf A}(2e_{2})=Id$, it follows that ${\bf A}$ is a spectrally unitary and an easy computation shows that ${\rm Fix}({\bf A})=\{0\}$. In fact, the Theorem 2 states that, for an adequate basis of $\mathbb{Z}^{p}$; any $\mathbb{Z}^{p}$-action on $\mathbb{Z}^{3}$ is essentially the action ${\bf A}$. For this, we need to show the following lemmas.

Lemma 2.2

Let $N$ and $M$ be in ${\rm GL}(3,\mathbb{Z})$ satisfying the following properties:

1. i.

$NM=MN$** 2. ii.

$1$* is an eigenvalue of $N^{n}\dot{M}^{m}$ for all $(n,m)\in\mathbb{Z}^{2}$.* 3. iii.

$\{k\in\mathbb{Z}^{3}/N^{n}\dot{M}^{m}k=k,\mbox{ for all }(n,m)\in\mathbb{Z}^{2}\}=\{0\}$.

Then after conjugation by an element of ${\rm GL}(3,\mathbb{Z})$, $N$ and $M$ have the following form

[TABLE]

In particular, the subgroup of ${\rm GL}(3,\mathbb{Z})$ generated by $N$ and $M$ is isomorphic to the Klein four-group.

Remark 2.3

Lemma 2.2 shows that the image of a spectrally unitary $\mathbb{Z}^{2}$-action on $\mathbb{Z}^{3}$ with trivial fixed point set is isomorphic to the Klein four-group. The following example shows that this is not true for a general spectrally unitary $\mathbb{Z}^{2}$-action on $\mathbb{Z}^{q}$ with trivial fixed point set.

Example 2.4

Let $\bf{A}$ be the action generated in the canonical basis $\{e_{1},e_{2}\}$ of $\mathbb{Z}^{2}$ by

[TABLE]

with $a\in\mathbb{Z}$. It is easy to see that $\bf{A}$ is a spectrally unitary $\mathbb{Z}^{2}$-action on $\mathbb{Z}^{4}$, satisfying the hypothesis of Lemma 2.2. Notice that the cyclic subgroup generated by ${\bf A}(e_{1})$ is not finite. In particular, the image of ${\bf A}$ is not isomorphic to the Klein four-group.

Proof of Lemma 2.2. Let $N$ and $M$ be matrices in ${\rm GL}(3,{\mathbb{C}})$ satisfying the hypothesis i, ii, and iii. The subspaces ${\rm Fix}(N)$, ${\rm Fix}(M)$, and ${\rm Fix}(MN)$ are non trivial subspaces (because of condition ii) whose pairwise intersections are reduced to $\{0\}$ (because of conditions i and iii). The direct sum ${\rm Fix}(N)\oplus{\rm Fix}(M)$ has dimension of at least two and intersects trivially ${\rm Fix}(NM)$ because each subspace ${\rm Fix}(N)$ and ${\rm Fix}(M)$ is invariant by $NM$ (because of i). We can deduce that ${\rm Fix}(MN)$ has dimension one, that ${\rm Fix}(N)\oplus{\rm Fix}(M)$ has dimension two (which of course implies that each space ${\rm Fix}(N)$, ${\rm Fix}(M)$ has dimension one), and

[TABLE]

Each of these three subspaces being invariant by both $M$ and $N$, these two matrices are simultaneously diagonalizable.

Let $a(N)$, $b(N)$, $c(N)$, $a(M)$, $b(M)$, and $c(M)$ be the diagonal coefficients of $N$ and $M$, respectively. We can suppose that $a(N)=1$, $b(N)\not=1$, $c(N)\not=1$, $a(M)\not=1$, $b(M)=1$, and $c(M)=c(N)^{-1}\not=1$. We observe that for every pair of integer $n$ and $m$, one of the numbers $b(N)^{n}$, $a(M)^{m}$ and $c(N)^{n}c(M)^{m}=c(N)^{n-m}$ is equal to 1. Setting $n=2$ and $m=1$, one gets $b(N)=-1$; setting $n=1$ and $m=2$, one gets $a(M)=-1$, setting $n=1$ and $m=-1$, one gets $c(N)=-1$. This implies that $M$ and $N$ generate the Klein four-group.

Now, we show that there exists a basis of $\mathbb{Z}^{3}$ such that in this basis, $N$ and $M$, have the desired form. In fact, we can assume by  [4] that there exists a basis $\{e_{1},e_{2},e_{3}\}$ of $\mathbb{Z}^{3}$ such that

[TABLE]

In particular, $\tilde{a}_{23}=0$ since $N^{2}=I$, where $I$ is the identity matrix. Considering that the eigenspace of $N$ associated with the eigenvalue $1$ has dimension one and $NMe_{1}=Me_{1}$, we have $Me_{1}=me_{1}$, and therefore $m$ is an eigenvalue of $M$. Hence, $m=\pm 1$. Thus from hypothesis $(iii)$ we have that $m=-1$. Then, on this basis, $M$ has the form

[TABLE]

Let B=\left[\begin{array}[]{cc}\tilde{b}_{22}&\tilde{b}_{23}\\ \tilde{b}_{32}&\tilde{b}_{33}\end{array}\right]. Then $\det M=-\det B=1$. Hence $\det B=-1$. Since $-1$ is an eigenvalue of multiplicity two, we have that $-1$ is an eigenvalue of $B$. Then it is easy to see that there exists a matrix $U\in{\rm GL}(2,\mathbb{Z})$ such that

[TABLE]

Therefore

[TABLE]

and

[TABLE]

Since $\det M=1$, we have $b_{33}=1$. Now, from hypothesis $(i)$, we have $b_{12}=0$ and $a_{12}b_{23}=-2(a_{13}+b_{13})$, thus finishing the proof.

$\hfill\square$

An immediate consequence of Lemma 2.2 is the following result.

Lemma 2.5

Let $N$ and $M$ be in ${\rm GL}(3,\mathbb{Z})$ under the same hypothesis of Lemma 2.2 and $C\in{\rm GL}(3,\mathbb{Z})$. If $C$ commutes with $N$ and $M$, and $1$ is an eigenvalue of $N^{n}M^{m}C^{s}$ for all $(n,m,s)\in\mathbb{Z}^{3}$, then $C$ belongs to the group generated by $N$ and $M$.

Proof. By Lemma 2.2 we observe that the decomposition

[TABLE]

is invariant by every element $C$ in ${\rm GL}(3,{\mathbb{C}})$ that commutes with $M$ and $N$. Writing $a(C)$, $b(C)$, and $c(C)$ the diagonal coefficients of $C$, we can deduces that if the hypothesis of Lemma 2.5 is satisfied, then for every integer $n$,$m$, $s$, one of the coefficient $(-1)^{m}a(C)^{s}$, $(-1)^{n}b(C)^{s}$, $(-1)^{n+m}c(C)^{s}$ is equal to 1. It is straight forward that $C$ is equal to $I$, $M$, $N$, or $MN$.

$\hfill\square$

Remark 2.6

Lemma 2.5 states that the abelian group $\left<N,M\right>$ generated by $N$ and $M$ is maximal in ${\rm GL}(3,\mathbb{Z})$ in the following sense: if ${\rm G}$ is an abelian subgroup of ${\rm GL}(3,\mathbb{Z})$ that contains $N$ and $M$, and $1$ is an eigenvalue of each element of ${\rm G}$, then ${\rm G}=\left<N,M\right>$.

Let $p$ be a natural number. For each $i=1,2,...,p$, we denote by ${\rm G}_{i}$ the subgroup of $\mathbb{Z}^{p}$ defined by ${\rm G}_{i}=\{(a_{1},a_{2},...,a_{p})\in\mathbb{Z}^{p}\mid a_{i}=0\}$.

Given a $\mathbb{Z}^{p}$-action ${\bf A}$ on $H_{1}(\mathbb{T}^{3},\mathbb{Z})$ acting by automorphisms of $\mathbb{Z}^{3}$, we shall denote by ${\bf A}_{i}$ the restriction of the action ${\bf A}$ to the subgroup $G_{i}$. If $\{e_{1},...,e_{p}\}$ is the canonical basis of $\mathbb{Z}^{p}$, it is easy to see that ${\rm Fix}({\bf A}_{i})=\cap_{j\neq i}{\rm Fix}({\bf A}(e_{j}))$, for all $i$. Thus we obtain the following lemma.

Lemma 2.7

Let ${\bf A}$ be an spectrally unitary $\mathbb{Z}^{p}$-action on $\mathbb{Z}^{3}$. If $p\geq 3$ and ${\rm Fix}({\bf A})=\{0\}$, then ${\rm Fix}({\bf A}_{i})=\{0\}$, for some $i$ such that $1\leq i\leq p$.

Proof. Suppose the lemma is false. Then there exists a set of nonzero vectors $\{v_{1},v_{2},...,v_{p}\}\subset\mathbb{Z}^{3}$ such that $v_{i}\in{\rm Fix}({\bf A}_{i})$ for all $i=1,2,...,p$.

Assume $\{v_{1},v_{2},...,v_{p}\}$ is linearly dependent. Then there exist a natural number $k$, $k\leq p$, and rational numbers $\lambda_{1}$, $\lambda_{2}$,…,$\lambda_{p}$ such that

[TABLE]

It follows that ${\bf A}(e_{k})v_{k}=v_{k}$ and $v_{k}\in{\rm Fix}({\bf A})$. Therefore ${\rm Fix}({\bf A})\neq\{0\}$, a contradiction with the hypothesis.

Now suppose $\{v_{1},v_{2},...,v_{p}\}$ is linearly independent. Then $p=3$ and there exist rational numbers $a_{11},a_{21}$, and $a_{31}$ such that ${\bf A}(e_{1})v_{1}=a_{11}v_{1}+a_{21}v_{2}+a_{31}v_{3}$. Since ${\rm Fix}({\bf A})=\{0\}$ and ${\bf A}(e_{1})v_{1}$ is invariant by ${\bf A}(e_{k})$ for $k=2,3$, we have $a_{11}=-1$ and $a_{21}=a_{31}=0$. Therefore ${\bf A}(e_{1})v_{1}=-v_{1}$. In a similar way we prove that ${\bf A}(e_{2})v_{2}=-v_{2}$ and ${\bf A}(e_{3})v_{3}=-v_{3}$. Thus, $1$ is not an eigenvalue of ${\bf A}(e_{1}){\bf A}(e_{2}){\bf A}(e_{3})$, which contradicts the hypothesis. $\hfill\square$

Proof of Theorem 2. We proceed by induction on $p$. If $p=2$, we may assume $A(e_{1})$ and $A(e_{2})$ satisfy the hypothesis of Lemma 2.2. Hence, after conjugation by an element of ${\rm GL}(3,\mathbb{Z})$, we may assume that $A(e_{1})$ and $A(e_{2})$ are in the normal form $(\ref{nor})$. Thus, the canonical basis $\{e_{1},e_{2}\}$ of $\mathbb{Z}^{2}$ has the required properties.

Assume the theorem holds for $p-1$; we will prove that it also holds for $p$. Since ${\rm Fix}({\bf A})=\{0\}$, by Lemma 2.7 we see that ${\rm Fix}({\bf A}_{i})=\{0\}$ for some $i=1,2,...,p$. Without loss of generality, we can assume that $i=p$. Then by the induction hypothesis there exists a basis $\{v_{1},...,v_{p-1}\}$ of $G_{p}$ such that $A(v_{1})$ and $A(v_{2})$ are in the normal form by $(\ref{nor})$ and $A(v_{j})=I$ for all $j>2$. Since $A(e_{p})$ commutes with $A(v_{1})$ and $A(v_{2})$ then from Lemma 2.5 we have $A(e_{p})=A(v)$ for some $v\in\{0,v_{1},v_{2},v_{1}+v_{2}\}$. So the basis $\{w_{1},w_{2},...,w_{p-1},w_{p}\}$ of $\mathbb{Z}^{p}$, where $w_{k}=v_{k}$ for all $k<p$ and $w_{p}=v+e_{p}$, has the required properties, proving Theorem 2. $\hfill\square$

3 Proof of Theorem 1

To prove Theorem 1, we use the following proposition.

Proposition 3.1

Let $G$ be a torsion-free group and $\varphi$ a $G$-action on a set $X$. Let $H$ be a normal subgroup of $G$ of finite index. Then $\varphi$ is a free $G$-action if and only if the restriction of the action $\varphi$ to $H$ is free.

Proof. If $\varphi$ is a free $G$-action then any restriction of $\varphi$ is free. Conversely, suppose there exists $g\in G$ such that $\varphi(g)$ has a fixed point $x\in X.$ Let $m=\left[G:H\right]$. Then $g^{m}\in H$ and $\varphi(g^{m})x=\varphi^{m}(g)x=x$. Since the restriction of $\varphi$ to $H$ is free then $g^{m}=e,$ where $e$ is the identity element of $G$. Since $G$ is a torsion-free group, we have that $g=e$. $\hfill\square$

Suppose ${\rm Fix}({\bf A})=\{0\}$. By the normal form of the action ${\bf A}$ given in Theorem 2, there exists a basis $\{w_{1},...,w_{p}\}$ of $\mathbb{Z}^{p}$ such that $A(w_{1})=N$, $A(w_{2})=M$, and $A(w_{j})=I$ for $j>2$, where the matrices $N$ and $M$ are defined as in Lemma 2.2. Consider the diffeomorphisms $\varphi_{i}$, $i=1,...,p$ of the torus $\mathbb{T}^{3}$ whose linear part is determined by the matrix $A(w_{i})$, $i=1,...,p$ respectively, and which are given on the covering $\mathbb{R}^{3}$ by

[TABLE]

where $f_{i}$ and $g_{j}$, with $i,j=1,2,...,p$, are $\mathbb{Z}$- periodic functions of the variable $z$ with Lebesgue measure zero and $r$ is a rational number. The family of diffeomorphisms $\widetilde{\varphi}_{i}$ defines a $\mathbb{Z}^{p}$-action on the torus $\mathbb{T}^{3}$ if and only if

[TABLE]

for all $i,j=1,2,...,p$.

Since the equations in $(\ref{eq7})$ are trivially satisfied for $i\geq 3$ and $j\geq 3$, we need to verify the equations in $(\ref{eq7})$ for $i=1,2$ and $j=1,2,3,...,p$. Let $r=-\tfrac{b}{4}$, $R(z)=-z$, and $T_{\tfrac{1}{2}}(z)=z+\tfrac{1}{2}$. First, we consider the case $i=1$ and $j=2$, then we have

[TABLE]

[TABLE]

It is easy to see that $\widetilde{\varphi}_{2}$ and $\widetilde{\varphi}_{1}$ satisfy the equation $(\ref{eq7})$, if and only if

[TABLE]

When $i=1$ and $j\geq 3$ we have

[TABLE]

and

[TABLE]

Thus the condition $(\ref{eq7})$ is verified if

[TABLE]

In the case where $i=2$ and $j\geq 3$ we have

[TABLE]

and

[TABLE]

In this case, condition $(\ref{eq7})$ is satisfied if

[TABLE]

We choose $f_{1}\left(z\right)=\tfrac{\alpha_{1}}{2}\cos 2\pi z$, $g_{1}\left(z\right)=-\tfrac{\alpha_{1}}{2}\sin 2\pi z$, $f_{2}\left(z\right)=-\tfrac{\alpha_{2}}{2}\cos 2\pi z+\tfrac{a}{4}\alpha_{2}\sin 2\pi z$, $g_{2}\left(z\right)=-\tfrac{\alpha_{2}}{2}\sin 2\pi z$, and for $j>2$ we choose $f_{j}\left(z\right)=\alpha_{j}(\cos 2\pi z-\tfrac{a}{2}\sin 2\pi z)$ and $g_{j}\left(z\right)=-\alpha_{j}\sin 2\pi z$, where $\alpha_{1},\alpha_{2},...,\alpha_{p}$ are arbitrary real numbers. A straightforward calculation shows that equations $(\ref{eqf12})$, $(\ref{eqf1j})$, and $(\ref{eqg2j})$ are verified. Thus the family of diffeomorphisms $\widetilde{\varphi}_{j}$, with $1\leq j\leq p$, defines a $\mathbb{Z}^{p}$-action on the torus $\mathbb{T}^{3}$. In what follows, we assume that the real numbers $\alpha_{1},\alpha_{2},...,\alpha_{p}$ are linearly independent over the field of algebraic numbers. We will show that the $\mathbb{Z}^{p}$-action $\varphi$ defined by the family $\widetilde{\varphi}_{j}$ is a free $\mathbb{Z}^{p}$-action. For this, by Proposition 3.1 it suffices to show that the restriction of $\varphi$ to the subgroup ${\rm H}=\{2\ell_{1}w_{1}+2\ell_{2}w_{2}+...+\ell_{p}w_{p}|\ell_{1},...,\ell_{p}\in\mathbb{Z}\}$ is free. From $(\ref{varhi1})$ and $(\ref{varhi2})$ we have that

[TABLE]

and

[TABLE]

thus we see that

[TABLE]

[TABLE]

[TABLE]

If

[TABLE]

has a fixed point $\pi(x,y,z)\in\mathbb{T}^{3}$, then, on the covering $\mathbb{R}^{3}$, there exists $(n_{1},n_{2},n_{3})\in\mathbb{Z}^{3}$ such that

[TABLE]

From $(\ref{eqpar})$ we see that

[TABLE]

and

[TABLE]

Replacing $(\ref{eqsin})$ in equation $(\ref{eqcos})$ we get

[TABLE]

From $(\ref{eqsin})$ and $(\ref{eqcos1})$ we see that

[TABLE]

Since the family $\{\alpha_{j}|j=1,2,...,p\}$ was chosen to be linearly independent over the field of algebraic numbers, we have $\ell_{1}=\ell_{2}=...=\ell_{p}=0$. Thus, Theorem 1 is proved.

$\hfill\square$

4 Comments.

Since the strong question is false in general, we can reformulate this question for $\mathbb{Z}^{p}$-actions on the torus $T^{q}$ that preserve the Haar measure and with its corresponding affine action conjugated to its linear part.

Let $\varphi$ be a $\mathbb{Z}^{p}$-action on the torus $\mathbb{T}^{q}$ and ${\bf A}$ be the induced action on $H_{1}(\mathbb{T}^{q},\mathbb{Z})$. Consider for each $\ell\in\mathbb{Z}^{p}$ a lifting $\widetilde{\varphi}(\ell):\mathbb{R}^{q}\rightarrow\mathbb{R}^{q}$ on the covering $\pi:\mathbb{R}^{q}\rightarrow T^{q}$ of $\varphi(\ell):\mathbb{T}^{q}\rightarrow\mathbb{T}^{q}$. Thus $\widetilde{\varphi}(\ell)$ can be written as

[TABLE]

where $\widetilde{F}(\ell)=F(\ell)\circ\pi$ and $F(\ell):\mathbb{T}^{q}\rightarrow\mathbb{R}^{q}$.

Denote by $a$ the $\mathbb{Z}^{p}$-action on $\mathbb{T}^{q}$ which acts by automorphisms of $\mathbb{T}^{q}$, defined by the action ${\bf A}$, i.e.

[TABLE]

for all $\ell\in\mathbb{Z}^{p}.$

An application $\alpha:\mathbb{Z}^{p}\rightarrow\mathbb{T}^{q}$ is 1-cocycle over the $\mathbb{Z}^{p}$-action $a$ if for each $\ell$, $\ell^{\prime}$ $\in\mathbb{Z}^{p}$

[TABLE]

A 1-cocycle is trivial if there exist a point $x_{0}\in\mathbb{T}^{q}$ such that

[TABLE]

for all $\ell\in$ $\mathbb{Z}^{p}.$ A leading example of 1-cocycle over $a$ is given by invariant probability measures of the action. Indeed, if $\mu$ is an invariant probability measure for $\varphi$, then the application $\alpha_{\mu}$ defined by

[TABLE]

for all $\ell\in$ $\mathbb{Z}^{p}$, where $F$ is as in $(\ref{ques})$, is a 1-cocycle over $a$.

Now, let $\varphi$ be a $\mathbb{Z}^{p}$-action on $\mathbb{T}^{q}$ that preserves the Haar measure $m$ of $\mathbb{T}^{q}$. Suppose that ${\rm Fix}({\bf A})=\{0\}$ and that the 1-cocycle $\alpha_{m}$ is trivial. We propose the following question: does the $\varphi$ action has a finite orbit?

The theorem 1 gives a negative answer to the stronger question in  [6], but the 1-cocycle induced by the Haar measure is not trivial.

Acknowledgments.

The authors want to thank Juan Rivera Letelier and Nathan M. dos Santos for helpful discussions.