Generalized torsions in once punctured torus bundles

A group $\Gamma$ is called a bi-orderable group if it admits total order $<$ which is invariant under multiplication to both sides, i.e. if $g<g^{\prime}$ then $fgh<fg^{\prime}h$ for every $f,h\in\Gamma$. As a convention, the trivial group is regarded as a bi-orderable group. There is one obstruction for groups being bi-orderable, called generalized torsions. An non-trivial element $g\in\Gamma$ is called generalized torsion if some product of conjugates of $g$ is the identity of $\Gamma$. In general, there exist groups which are not bi-orderable and have no generalized torsions [2], [3], [10]. However, it is conjectured that bi-orderability and having no generalized torsions is equivalent for the class of 3-manifold groups [8]. We have many affirmative answers for this conjecture, for some geometric manifolds, for some manifolds with non-trivial geometric decompositions, for many link complements and their Dehn fillings [5],[6],[8],[9],[11],[14],[15].

For the complements of fibered links (not necessarily in $S^{3}$), the following important sufficient condition and necessary condition (not necessary and sufficient condition) for the bi-orderability are known.:

In this paper, we consider once punctured torus bundles, surface-bundles over circles each of whose fiber is torus with connected boundary. In this case, the conditions above are necessary and sufficient condition for the bi-orderability. We will show that if the fundamental group of a once punctured torus bundle is not bi-orderable then it admits a generalized torsion. This adds the class of once punctured torus bundles to the list of 3-manifolds where the conjecture holds.

The rest of this paper is organized as follows. In Section 2, we represent the fundamental groups as oriented based paths and the conjugation classes of the fundamental groups as oriented free loops. We review the relations between theses and some operations. In Section 3, we review that generalized torsions in the fundamental group of fiber-bundles over circles are in that of the fibers and satisfy some relation in terms of monodromies. In Section 4, we assign words and cyclic words to oriented based paths and oriented free loops in once punctured torus, respectively. After these preparations, we prove the main result in Section 5. Thanks to Theorem 1.1 and Theorem 1.2, we know when the fundamental group of a once punctured torus bundle is not bi-ordearble. We will find a generalized torsion in this case. At last, as a remark, we show that our result contains a generalized torsion in the fundamental group of a tunnel number two hyperbolic once punctured torus bundle.

In this paper, we do not distinguish the homotopy classes of paths or loops with their representatives. The words in groups are read from left to right. For a group $\Gamma$, $C(\Gamma)$ denotes the set of conjugacy classes of $\Gamma$.

The author would like to thank professor Motegi for introducing the area of generalized torsions to him, and giving him many helpful comments.

For a topological space $X$ and a base point $*\in X$, the fundamental group $\pi_{1}(X,*)$ is defined as the set of the homotopy types of oriented based paths $([0,1],\{0,1\})\longrightarrow(X,*)$ with multiplications as concatenations. Let $F:\pi_{1}(X,*)\longrightarrow C\left(\pi_{1}(X,*)\right)$ be a map which sends an element to its conjugacy class. Topologically, $C\left(\pi_{1}(X,*)\right)$ is regarded as the set of the homotopy types of oriented free loops $S^{1}\longrightarrow X$ and $F$ as the operation which identifies the endpoints of paths and forgets the identified special point.

In this section, we show that generalized torsions in the fundamental group of a fiber bundle over a circle are same as the elements in the fundamental group of the fiber space satisfying some relations (Proposition 3.1).

Let $(X,*)$ be a based topological space and $\phi$ a self-homeomorphism of $X$ fixing $*$. Let $(Y,*_{Y})$ be a topological space obtained from $X\times[0,1]$ by identifying $(x,1)$ and $(\phi(x),0)$ for $x\in X$, and $*_{Y}$ comes from $(*,0)$. In this situation, $(Y,*_{Y})$ is called the $(X,*)$-bundle over a circle using $\phi$ and $\phi$ is called monodromy. Let $\tau\in\pi_{1}(Y,*_{Y})$ be the based path coming from $[0,1]\longrightarrow X\times[0,1];t\longmapsto(*,t)$ and let $T$ be a subgroup of $\pi_{1}(Y,*_{Y})$ generated by $\tau$. Then it is known that $T$ is isomorphic to $\mathbb{Z}$, and that $\pi_{1}(X,*)$ injects in $\pi_{1}(Y,*_{Y})$. Under these notation, $\pi_{1}(Y,*_{Y})$ is isomorphic to $\pi_{1}(X,*)\rtimes T$, where the multiplication is defined as $(g_{1},\tau^{n_{1}})\cdot(g_{2},\tau^{n_{2}})=(g_{1}\tilde{\phi}^{n_{1}}(g_{2}),\tau^{n_{1}+n_{2}})$, where $\tilde{\phi}$ is the induced map between $\pi_{1}(X,*)$ by $\phi$. Abusing notations, $(g,1_{T})$ for $g\in\pi_{1}(X,*)$ and $(1_{\pi_{1}(X,*)},\tau^{n})$ are written as $g$ and $\tau^{n}$, respectively. Note that for $g\in\pi_{1}(X,*)$, we have $\tau g\tau^{-1}=\tilde{\phi}(g)$ and $\tau^{-1}g\tau=\tilde{\phi}^{-1}(g)$ in $\pi_{1}(Y,*_{Y})$.

Suppose that $g\in\pi_{1}(Y,*_{Y})$ is a generalized torsion. Take positive integer $n$ and $w^{\prime}_{1},\dots,w^{\prime}_{n}\in\pi_{1}(Y,*_{Y})$ such that $(w^{\prime}_{1}g{w^{\prime}_{1}}^{-1})\cdots(w^{\prime}_{n}g{w^{\prime}_{n}}^{-1})=1_{\pi_{1}(Y,*_{Y})}$. By considering the projection $\pi_{1}(Y,*_{Y})\longrightarrow T$, we have $g\in\pi_{1}(X,*)$. Represent $w^{\prime}_{i}$ as $g_{i_{1}}\tau^{n_{i_{1}}}\cdots g_{i_{k}}\tau^{n_{i_{k}}}$, where $g_{i_{j}}\in\pi_{1}(X,*)$ and $n_{i_{j}}\in\mathbb{Z}$ for $j=1,\dots k$. Then $(w^{\prime}_{i}g{w^{\prime}_{i}}^{-1})=\left(\prod^{k}_{j=1}\tilde{\phi}^{\left(\sum^{j}_{l=1}n_{i_{l}}\right)}(g_{i_{j}})\right)\tilde{\phi}^{\left(\sum^{k}_{l=1}n_{i_{l}}\right)}(g)\left(\prod^{k}_{j=1}\tilde{\phi}^{\left(\sum^{j}_{l=1}n_{i_{l}}\right)}(g_{i_{j}})\right)^{-1}$ in $\pi_{1}(Y,*_{Y})$. Set $N_{i}$ to be $\sum^{k}_{l=1}n_{i_{l}}$ and $w_{i}\in\pi_{1}(X,*)$ to be $\left(\prod^{k}_{j=1}\tilde{\phi}^{\left(\sum^{j}_{l=1}n_{i_{l}}\right)}(g_{i_{j}})\right)$. Then we have an equation $\left(w_{1}\tilde{\phi}^{N_{1}}(g){w_{1}}^{-1}\right)\cdots\left(w_{n}\tilde{\phi}^{N_{n}}(g){w_{n}}^{-1}\right)=1_{\pi_{1}(Y,*_{Y})}$ in $\pi_{1}(Y,*_{Y})$. Since the left-hand side is a composition of elements in $\pi_{1}(X,*)$ and $\pi_{1}(X,*)$ injects in $\pi_{1}(Y,*_{Y})$, we have an equation $\left(w_{1}\tilde{\phi}^{N_{1}}(g){w_{1}}^{-1}\right)\cdots\left(w_{n}\tilde{\phi}^{N_{n}}(g){w_{n}}^{-1}\right)=1_{\pi_{1}(X,*)}$ in $\pi_{1}(X,*)$.

Conversely, suppose that there exist $g\in\pi_{1}(X,*)$, positive integer $n$, integers $N_{1},\dots,N_{n}$ and elements $w_{1}\dots,w_{n}\in\pi_{1}(X,*)$ such that $\left(w_{1}\tilde{\phi}^{N_{1}}(g){w_{1}}^{-1}\right)\cdots\left(w_{n}\tilde{\phi}^{N_{n}}(g){w_{n}}^{-1}\right)=1_{\pi_{1}(X,*)}$ holds in $\pi_{1}(X,*)$. Then $\left(w_{1}\tau^{N_{1}}g\tau^{-N_{1}}{w_{1}}^{-1}\right)\cdots\left(w_{n}\tau^{N_{n}}g\tau^{-N_{n}}{w_{n}}^{-1}\right)=1_{\pi_{1}(Y,*_{Y})}$ holds in $\pi_{1}(Y,*_{Y})$. This implies that $g$ is a generalized torsion.

We state the above argument as a Proposition:

Let $(\Sigma,*)$ be a torus with connected boundary with a base point on the boundary. In this section, we review the way to represent oriented based paths, elements of $\pi_{1}(\Sigma,*)$, and oriented free loops, elements of $C\left(\pi_{1}(\Sigma,*)\right)$. Take two disjoint properly embedded oriented arcs $I_{x}$ and $I_{y}$ such that they are disjoint from $*$ and cut $\Sigma$ into a disk as in Figure 2.

Let $(\Sigma,*)$ be a torus with connected boundary with a base point on the boundary, and take the same arcs $I_{x}$ and $I_{y}$ as in Figure 2 and the same loops $\alpha$ and $\beta$ as in Figure 3. Let $(M_{\phi},*_{M_{\phi}})$ be the $\Sigma$-bundle over a circle with orientation preserving and boundary fixing monodromy $\phi$. The 3-manifolds admitting such structures are called once punctured torus bundles, (OPTB for short). The representation matrix under the basis $\{[\alpha],[\beta]\}$, which are the homology classes of $\alpha$ and $\beta$, of the induced isomorphism of $H_{1}(\Sigma;\mathbb{Z})$ by $\phi$ is denoted by $A_{\phi}\in SL_{2}(\mathbb{Z})$. Then the following Fact are known:

Since the eigenvalues of $A\in SL_{2}(\mathbb{Z})$ is the roots of $x^{2}-(trA)x+1=0$, Theorem 1.1 implies that $\pi_{1}(M_{\phi},*_{M_{\phi}})$ is bi-orderable (and thus admits no generalized torsions) if $trA_{\phi}\geq 2$. Moreover, Theorem 1.2 implies that $\pi_{1}(M_{\phi},*_{M_{\phi}})$ is not bi-orderable if $trA_{\phi}<2$. We want to show that $\pi_{1}(M_{\phi},*_{M_{\phi}})$ admits a generalized torsion if $trA_{\phi}<2$.

Under Fact 5.1, we assume that representing matrix $A_{\phi}$ is of the form of the following:

Hereafter, we assume that $A_{\phi}=\left(\begin{array}[]{cc}a&b\\ c&d\\ \end{array}\right)$ with $ad\geq 0$ and $|a|\geq|d|$. Note that $\alpha\left(=L(1,0)\right)$ is mapped to $L(a,c)$ by $\phi$.