# Non-standard bi-orders on punctured torus bundles

Jonathan Johnson and Henry Segerman

###### Abstract.

Results of Perron and Rolfsen imply that untwisted hyperbolic once-punctured
torus bundles over the circle have bi-orderable fundamental groups. They do
this by showing that the action of the monodromy preserves a “standard” bi-
ordering formed using the lower central series of the free group. Here we
investigate other bi-orderings that punctured torus bundle groups can have. We
show that for every such bi-ordering, the largest and second largest proper
convex subgroups match the corresponding convex subgroups for a standard bi-
ordering. Moreover, if there exists a third largest convex subgroup, it must
also match the third largest convex subgroup for a standard bi-ordering.
However, we also show that these groups admit non-standard bi-orderings.

##  1\. Introduction

Much progress has been made on determining when a three-manifold group is
_left-orderable_ , that is, having a strict total order invariant under left
(but not necessarily right) multiplication. In fact, there is a conjectured
answer to when a three-manifold group is left-orderable [1]. However,
comparatively little is known about the symmetrized version of this property.

######  Definition 1.1.

A group is _bi-orderable_ if there is a strict total order of its elements
invariant under left and right multiplication. ∎

Most work on bi-orderability to date has focused on the question of which
three-manifolds have bi-orderable fundamental groups. Boyer, Rolfsen, and
Wiest [2] classified all Seifert fibered spaces with bi-orderable fundamental
groups. Perron and Rolfsen [17] proved a sufficient condition for bi-
orderability of the groups of fiber bundles over the circle. Later, Clay and
Rolfsen [4] proved an obstruction for bi-orderability of these fiber bundle
groups. Linnell, Rhemtulla, and Rolfsen [15] generalized Perron and Rolfsen’s
result, in particular to address some non-fibered manifolds. These results
have been used to show that many link complements have bi-orderable groups [3,
11, 12, 13, 18].

In this paper we turn from asking which three-manifolds have bi-orderable
fundamental groups to the question of classifying the bi-orders on a bi-
orderable fundamental group. In particular, we look at punctured torus
bundles. Clay, Perron, and Rolfsen’s results [4, 17] together determine that
the punctured torus bundles with bi-orderable fundamental groups are the
untwisted hyperbolic punctured torus bundles (Proposition 3.2). Thus for the
rest of the paper we will assume that our punctured torus bundles are of this
form.

The following notions to do with _convexity_ are key to describing the
structure of bi-ordered groups.

######  Definition 1.2.

A subgroup $C$ of a bi-ordered group $(G,<)$ is _convex with respect to $<$_
if for every $a,b\in C$ and $g\in G$, when $a<g<b$ then $g\in C$. We call a
convex subgroup $C$ _maximal_ if $C$ is a proper subgroup and the only convex
subgroups containing $C$ are $C$ and $G$. ∎

Our first result tells us that there is no choice in the maximal convex
subgroup for a bi-orderable hyperbolic punctured torus group. We can then take
that maximal convex subgroup and ask what _its_ maximal convex subgroup is.
Again there is no choice.

######  Theorem 1.3.

Let $M$ be an untwisted hyperbolic punctured torus bundle with fundamental
group

$\pi_{1}(M)\cong G\rtimes\mathbb{Z}$

where $G$ is the fundamental group of the punctured torus. Let $<$ be a bi-
ordering of $\pi_{1}(M)$.

  1. (1)

The maximal convex subgroup of $\pi_{1}(M)$ with respect to $<$ is $G$.

  2. (2)

Let $<_{G}$ be the induced bi-order of $<$ on $G$. The maximal convex subgroup
of $G$ with respect to $<_{G}$ is $G_{2}=[G,G]$.

A maximal convex subgroup is normal (Proposition 2.12), and it follows from a
theorem of Hölder that the quotient is abelian [10]. Thus the ‘‘most
significant digit’’ of any such bi-order is given by a bi-ordering of this
finitely generated free abelian group; such bi-orders are well understood.

One might ask if the pattern of maximal convex subgroups being determined
continues. Our next result tells us that if $G_{2}$ has a maximal convex
subgroup then, again, there is no choice.

######  Theorem 1.4.

Let $M$ be an untwisted hyperbolic punctured torus bundle with fundamental
group

$\pi_{1}(M)\cong G\rtimes_{h}\mathbb{Z}$

where $G$ is the fundamental group of the punctured torus. Let $<$ be a bi-
ordering of $\pi_{1}(M)$. Let $<_{2}$ be the induced bi-order of $<$ on
$G_{2}=[G,G]$. Suppose that $C$ is a maximal convex subgroup of $G_{2}$ with
respect to $<_{2}$. Then $C$ is $G_{3}=[G_{2},G]$.

In Theorem 1.3 the groups $\pi_{1}(M)$ and $G$ must have maximal convex
subgroups because they are finitely generated (Proposition 2.12). However,
$G_{2}$ is not finitely generated, leaving open the possibility that it fails
to have a maximal convex subgroup.

In Section 5 we build bi-orders on $\pi_{1}(M)$ for which $G_{2}$ does not
have a maximal convex subgroup (Corollary 5.10). Our examples are not
obtainable by existing construction methods in the literature. Perron and
Rolfsen [17] show that untwisted hyperbolic punctured torus bundles are bi-
orderable by showing that the monodromy of each of these bundles preserves at
least one bi-ordering of the free group. These bi-orderings are _standard_ ,
meaning that every term of the lower central series is convex (although each
term may not be maximal in the previous term); see Definition 2.22 and
Proposition 2.24 for the details. In a natural extension of this terminology,
we say that the fundamental group of a punctured torus bundle is _standard_ if
$h_{*}$ preserves a standard bi-ordering on the free group $G$.

######  Theorem 1.5.

Let $M$ be an untwisted hyperbolic punctured torus bundle. Then $\pi_{1}(M)$
admits a non-standard bi-ordering.

###  1.1. Bi-orders and eigenvalues

Here we give some relevant context to our results and discuss possible future
directions.

In this section, let $M$ be a fiber bundle over the circle whose fiber is a
once punctured surface $S$ (not necessarily a genus one surface), and let $h$
be the monodromy of this bundle. As in the genus one case, $\pi_{1}(M)$
factors into a semidirect product $\pi_{1}(M)\cong G\rtimes_{h_{*}}\mathbb{Z}$
where $G\cong\pi_{1}(S)$ is a free group of rank twice the genus of $S$. Thus
there is a natural map $\pi_{1}(M)\to\mathbb{Z}$ obtained by projecting
$G\rtimes_{h}\mathbb{Z}$ onto the $\mathbb{Z}$ factor. Therefore a bundle over
the circle has a canonical Alexander polynomial $\Delta_{M}(t)$. Perron and
Rolfsen, and Clay and Rolfsen prove the following.

######  Theorem 1.6 ([17], Theorem 1.1).

If all the roots of $\Delta_{M}(t)$ are real and positive then $\pi_{1}(M)$
admits a standard bi-ordering.

######  Theorem 1.7 ([4], Theorem 1.1).

If $\pi_{1}(M)$ is bi-orderable then $\Delta_{M}(t)$ has at least one real and
positive root.

######  Remark 1.8.

These results are stated for fibered knot complements but the proofs can be
applied to arbitrary fiber bundles over the circle without any alterations. ∎

Theorem 1.6 is a sufficient but not necessary condition for $\pi_{1}(M)$ to
admit a standard bi-ordering. However, to know whether or not $\pi_{1}(M)$
admits a standard bi-ordering we only need to investigate the induced map
$h_{+}\colon H_{1}(S;\mathbb{Z})\to H_{1}(S;\mathbb{Z})$. (See Proposition
2.25 for more details.)

######  Question 1.9.

Suppose that $S$ is a once-punctured surface, and suppose that $M$ and
$M^{\prime}$ are $S$ bundles over the circle with monodromies $h$ and
$h^{\prime}$ respectively. Is it possible that $\pi_{1}(M^{\prime})$ is bi-
orderable, $\pi_{1}(M)$ is not bi-orderable, but in homology we have that
$h_{+}=h_{+}^{\prime}$? ∎

######  Remark 1.10.

If we allow $S$ to have multiple punctures in Question 1.9 then the answer is
“yes”. In particular, let $\mathrm{br}\,(\beta)$ be the link in $S^{3}$ formed
by the closure of a braid $\beta$ along with the braid axis. Let $M$ and
$M^{\prime}$ be the exteriors of
$\mathrm{br}\,(\sigma_{1}\sigma_{2}\sigma_{1})$ and
$\mathrm{br}\,(\sigma_{1}\sigma_{2}\sigma_{1}^{-1})$ respectively. The
manifolds $M$ and $M^{\prime}$, pictured in Figure 1.1, are $S$ bundles over
the circle where $S$ is a 4-punctured sphere. For both bundles, the induced
maps $h_{+},h_{+}^{\prime}\colon H_{1}(S;\mathbb{Z})\to H_{1}(S;\mathbb{Z})$
are described by the following matrix:

$h_{+}=h_{+}^{\prime}=\mathopen{}\mathclose{{}\left(\begin{array}[]{ccc}0&0&1\\\
0&1&0\\\ 1&0&0\end{array}}\right)$

Kin and Rolfsen showed that $\pi_{1}(M)$ is bi-orderable [13, Theorem 4.10]
but $\pi_{1}(M^{\prime})$ is not bi-orderable [13, Proposition 4.4, Corollary
4.15]. ∎

(a) $M$ (b) $M^{\prime}$

Figure 1.1. Bundles over the circle

By Proposition 2.25, an example answering Question 1.9 in the affirmative
would have to involve a non-standard bi-ordering. Theorem 1.5 confirms that
non-standard bi-orderings exist, although in our examples $M$ is hyperbolic
and $S$ is of genus one, a setting in which the answer to Question 1.9 is
known to be ‘‘no’’. (See Proposition 3.2: $h_{+}$ determines whether or not
$M$ is untwisted.)

A negative answer to Question 1.9 would have to arise from restrictions that
the map on homology $h_{+}$ places on the non-standard bi-orderings of
$\pi_{1}(M)$. In our examples, $h_{+}$ puts severe restrictions on what is
possible. One might expect the same to be true in higher genus.

######  Question 1.11.

What other non-standard bi-orderings of a hyperbolic punctured torus (or other
surface) bundle group are there beyond those generated by our construction? ∎

###  1.2. Structure of the paper

We begin with an exposition of the interplay between bi-orderings of groups
and their convex subgroups in Section 2. In Section 3 we recall the definition
and some results about punctured torus bundles. Section 4 focuses on the
restrictions that the monodromy imposes on the types of bi-orderings which can
be put on the rank 2 free group. Theorems 1.3 and 1.4 are proved in Section
4.1 and Section 4.3 respectively. We describe non-standard bi-ordering of a
hyperbolic punctured torus bundle group in Section 5 and so prove Theorem 1.5.

###  1.3. Acknowledgments

The authors thank Danny Calegari and Neil Hoffman for helpful conversations.
The authors were supported in part by National Science Foundation grants
DMS-2213213 and DMS-2203993 respectively.

##  2\. Bi-Orders and Convex Subgroups

In this section, we state some preliminary facts about bi-orderings and convex
subgroups of bi-ordered groups. In particular, we explore the interplay
between the bi-orderings of groups $G$ and $H$ related by a surjective
homomorphism $f\colon G\to H$. For a general reference for many of these
results see Section 2.2 of Clay and Rolfsen’s book [5]. We start by stating an
alternative way to describe a bi-ordering of a group.

###  2.1. Positive Cones and Induced Orderings

While the most natural method to define a bi-ordering of a group is to specify
a total ordering, a bi-ordering is also specified by its set of positive
elements.

######  Definition 2.1.

A subset $P\subset G$ is a _(conjugate invariant) positive cone_ if it
satisfies the following:

  1. (1)

$P\cdot P\subset P$

  2. (2)

$G=P\sqcup P^{-1}\sqcup\\{1\\}$

  3. (3)

$g^{-1}Pg=P$ for all $g$ in $G$. ∎

A bi-order $<$ of $G$ corresponds to a unique positive cone by $P_{<}=\\{g\in
G\mid 1<g\\}$. In the other direction, a bi-order can be obtained from a
(conjugate invariant) positive cone $P$ by defining $a<_{P}b$ if and only if
$a^{-1}b\in P$. We use $(G,<)$, $(G,P)$, or $(G,<,P)$ to mean a group with bi-
order $<$ and/or its corresponding positive cone $P$.

######  Remark 2.2.

The term “positive cone” is often used to define the positive elements of a
left-ordering of a group. In this case, condition (3) is not required. The
conjugate invariant positive cones of a group are precisely the positive cones
associated to left-orders which also happen to be bi-orders. Since we only
deal with positive cones of bi-orders, all positive cones in this paper are
assumed to be conjugate invariant. ∎

######  Definition 2.3.

Let $(G,<_{G},P_{G})$ and $(H,<_{H},P_{H})$ be bi-ordered groups, and let
$f\colon G\to H$ be a surjective homomorphism. The bi-order of $H$ is _induced
by_ the bi-order of $G$ if either of the following equivalent conditions is
true:

  1. (1)

For all $a,b\in G$, $a<_{G}b$ implies $f(a)\leq_{H}f(b)$,

  2. (2)

$f(P_{G}-\ker f)=P_{H}$. ∎

######  Remark 2.4.

Given a surjective homomorphism $f\colon G\to H$, each bi-ordering of $G$ can
induce either zero or one bi-ordering of $H$. However, a given bi-ordering of
$H$ may be induced by many distinct bi-orderings of $G$. ∎

The following lemma constructs one such bi-order of $G$.

######  Lemma 2.5.

Suppose $f\colon G\to H$ is a surjective homomorphism of groups. Let $P_{H}$
be a positive cone of $H$ and let $P_{K}$ be a positive cone of $K=\ker f$
invariant under conjugation by elements of $G$. The set

$P=\\{g\in G:f(g)\in P_{H}\text{ or }g\in P_{K}\\}$

is a positive cone of $G$ which induces $P_{H}$.

###### Proof.

By construction, $f(P-\ker f)\subset P_{H}$, and since $f$ is surjective and
$1\notin P_{H}$, we have $P_{H}\subset f(P-\ker f)$. Therefore $f(P-\ker
f)=P_{H}$. It remains to show that $P$ is a positive cone. We check the
properties in Definition 2.1 as follows.

  1. (1)

Consider $x,y\in P$. If $x,y\in P_{K}$ then $xy\in P_{K}$ so $xy\in P$. If
$f(x)\in P_{H}$ then either $f(xy)=f(x)f(y)$ is a product of elements in
$P_{H}$ or $f(xy)=f(x)$. Therefore, $xy\in P$. Similarly, if $f(y)\in P_{H}$
then $xy\in P$. This proves Definition 2.1(1).

  2. (2)

Since $P_{H}$ is a positive cone, $f(1)=1\notin P_{H}$. Since $P_{K}$ is a
positive cone, $1\notin P_{K}$. Therefore, $1\notin P$.

Consider $x\in P$. We will show that $x^{-1}\notin P$. Suppose $f(x)\in
P_{H}$. Then, $f(x^{-1})\notin P_{H}$, and since $f(x)\neq 1$, $x$ is not in
$\ker f$ so $x^{-1}\notin P_{K}$. Therefore, $x^{-1}\notin P$.

Suppose $x\in P_{K}$. Then $x^{-1}\notin P_{K}$ since $P_{K}$ is a positive
cone. Since $x\in K$ and $P_{H}$ is a positive cone, $f(x^{-1})=1\notin
P_{H}$. Therefore, $x^{-1}\notin P$.

Since $1\notin P$ and $P\cap P^{-1}=\emptyset$, it follows that $G=P\sqcup
P^{-1}\sqcup\\{1\\}$. This proves Definition 2.1(2).

  3. (3)

Consider $x\in P$, and suppose $g\in G$. If $f(x)\in P_{H}$, then
$f(g^{-1}xg)=f(g)^{-1}f(x)g(x)\in P_{H}$ since $P_{H}$ is a positive cone. If
$x\in P_{K}$, then $g^{-1}xg\in P_{K}$ since $P_{K}$ is invariant under
conjugation by elements of $G$. Therefore, $g^{-1}xg\in P$ and we have
Definition 2.1(3). ∎

An application of Lemma 2.5 is the following result on semidirect products of
groups with the integers.

######  Proposition 2.6.

Let $G$ be a group and let $\phi\in\mathsf{Aut}(G)$. A semidirect product
$G\rtimes_{\phi}\mathbb{Z}$ is bi-orderable if and only if there is a bi-order
of $G$ invariant under $\phi$.

###### Proof.

First suppose that $G\rtimes_{\phi}\mathbb{Z}$ has a bi-order $<$. The bi-
order $<$ restricts to a bi-order $<_{G}$ on $G$. Since the action of $\phi$
is conjugation in $G\rtimes_{\phi}\mathbb{Z}$ it preserves $<_{G}$.

In the other direction, suppose that there is a bi-order of $G$ with positive
cone $P_{K}$ invariant under $\phi$. By an abuse of notation, we also denote
the corresponding subset of $G\rtimes_{\phi}\mathbb{Z}$ as $P_{K}$. We apply
Lemma 2.5 with $G\rtimes_{\phi}\mathbb{Z}$ playing the role of $G$, the
integers $\mathbb{Z}$ playing the role of $H$, and the projection map $f\colon
G\rtimes_{\phi}\mathbb{Z}\to\mathbb{Z}$ being the surjective homomorphism. For
the positive cone $P_{H}$ on $\mathbb{Z}$ we take the set of positive
integers. We check that $P_{K}$ is invariant under conjugation by elements of
$G\rtimes_{\phi}\mathbb{Z}$ as follows. Consider an element $(x,0)\in P_{K}$
and an arbitrary element $(g,k)\in G\rtimes_{\phi}\mathbb{Z}$. We calculate

$(g,k)\cdot(x,0)\cdot(g,k)^{-1}=(g\phi^{k}(x)g^{-1},0)$

Since $P_{K}$ is invariant under $\phi$ and conjugation in $G$, we have that
$(g\phi^{k}(x)g^{-1},0)\in P_{K}$. Thus $P_{K}$ is invariant under conjugation
by elements of $G\rtimes_{\phi}\mathbb{Z}$. Applying Lemma 2.5, we get that

$P=\\{(g,n)\in G\rtimes_{\phi}\mathbb{Z}\mid 0<n\text{ or }(n=0\text{ and
}g\in P_{K})\\}$

is a positive cone of $G\rtimes_{\phi}\mathbb{Z}$. Therefore,
$G\rtimes_{\phi}\mathbb{Z}$ is bi-orderable. ∎

###  2.2. Convex Subgroups

Given a surjective homomorphism of bi-orderable groups $f\colon G\to H$, we
now consider when a bi-ordering of $G$ induces a bi-ordering of $H$. The
answer is related to the convex subgroups of $G$; see Definition 1.2. We first
restate the definition of convexity in terms of positive cones.

######  Proposition 2.7.

Let $(G,<,P)$ be a bi-ordered group. A subgroup $C\subset G$ is convex with
respect to $<$ if and only if for each $g\in G$, either $gC\subset P$,
$gC\subset P^{-1}$, or $gC=C$. The analogous statement for right cosets also
holds.

###### Proof.

Suppose that $C$ is convex with respect to $<$. Let $gC$ be a left coset of
$C$ such that $gC\neq C$. Thus, $g\notin C$. Let $h=gc$ where $c\in C$. For a
contradiction, assume that $g\in P$ and $h\in P^{-1}$. Therefore $h=gc<1<g$.
Multiplying by $g^{-1}$ on the left we get $c<g^{-1}<1$. Since $C$ is convex,
this implies that $g\in C$ and we have reached a contradiction. Therefore,
either $gC\subset P$ or $gC\subset P^{-1}$.

Now suppose that for each $g\in G$, either $gC\subset P$, $gC\subset P^{-1}$,
or $gC=C$. Let $a$ and $b$ be in $C$, and let $a<g<b$ for some $g\in G$. These
inequalities imply that $1<ga^{-1}$ and $gb^{-1}<1$. Thus, $gC$ is not a
subset of $P$ or $P^{-1}$. Therefore we have that $gC=C$ and so $g\in C$.

The proof for right cosets is similar. ∎

As a consequence of Proposition 2.7, we have the following.

######  Corollary 2.8.

Let $(G,<,P)$ be a bi-ordered group. If $C$ is a normal convex subgroup of $G$
then there is a bi-ordering on $G/C$ whose positive cone is the set of cosets
$gC$ such that $gC\subset P$. This bi-ordering of $G/C$ is the one induced by
$G$ using the quotient map. ∎

We now state two useful results about convex subgroups. We refer the reader to
a paper by Conrad [7] for proofs of these facts in the context of right-
orders. However, the arguments work equally well for bi-orders.

######  Proposition 2.9 (Conrad [7], Section 3.3).

Let $G$ be a bi-ordered group the set of all convex subgroups of $G$ is
totally ordered by inclusion. ∎

######  Remark 2.10.

Recall that a convex subgroup $C$ of a bi-ordered group $(G,<)$ is _maximal_
if $C$ is proper and the only convex subgroups containing $C$ are $G$ and $C$.
By Proposition 2.9, maximal convex subgroups are unique when they exist. ∎

######  Proposition 2.11 (Conrad [7], Section 3.6).

Let $(G,<)$ be a bi-ordered group. Suppose that $f\colon G\to H$ is a
surjective homomorphism of groups. If the bi-ordering $<$ induces a bi-
ordering on $H$ then $\ker f$ is convex with respect to $<$. In particular,
$H$ is bi-orderable if and only if $\ker f$ is convex with respect to some bi-
ordering of $G$. ∎

######  Proposition 2.12 (Clay and Rolfsen [4], Lemma 2.4).

If $(G,<)$ is a finitely-generated non-trivial bi-ordered group, then there
exists a unique maximal convex subgroup $C$ of $G$ satisfying $C\neq G$.
Moreover, $C$ is normal in $G$ and $G/C$ is abelian. If an automorphism
$\phi\colon G\to G$ preserves $<$ then $\phi(C)=C$. ∎

Given a bi-ordered group $(G,<)$, knowing information about the set of
subgroups convex with respect to $<$ can provide important information about
the bi-ordering $<$. The following theorem is a classical result of Hölder.
(See Clay and Rolfsen [5, Theorem 2.6] for a more recent exposition of this
result.)

######  Theorem 2.13 (Hölder [10]).

Suppose that $(G,<)$ is a bi-ordered group with no proper non-trivial convex
subgroups. Then there is an injective homomorphism $f\colon G\to\mathbb{R}$
such that the bi-order of the image of $f$ induced by $<$ is the same as the
one induced by the usual ordering of $\mathbb{R}$.

###  2.3. Constructing bi-orders

In this section we describe a general procedure for building bi-orders. Let
$G$ be a group, and consider a surjective homomorphism $f\colon G\to H$. Let
$C$ be the kernel of $f$. We can define a positive cone on $G$ by specifying a
positive cone of $H$ and a positive cone on $C$ invariant under conjugation by
elements of $G$ as in Lemma 2.5. By Proposition 2.11, the subgroup $C$ will be
convex with respect to this bi-order.

Abelian groups are a convenient choice for $H$ since bi-orderability is well
understood for these groups. In particular, Levi proves the following.

######  Theorem 2.14 (Levi [14], Section 3).

An abelian group is bi-orderable if and only if it is torsion-free. Thus any
free abelian group is bi-orderable. ∎

######  Example 2.15.

Suppose that $A\cong\mathbb{Z}^{2}$.

  1. (1)

One way to define a bi-ordering of $A$ is to compare elements one factor at a
time. In other words, an element $(m,n)\in A$ is positive when $m>0$ or when
$m=0$ and $n>0$. The set $\\{0\\}\times\mathbb{Z}$ is convex with respect to
this bi-ordering. This type of bi-ordering is referred to as a
_lexicographical bi-ordering_.

  2. (2)

Another option is to let $v\in\mathbb{R}\oplus\mathbb{R}$ be any vector with
irrational slope. Then (with the dot product defined as usual) the set
$P=\\{u\in A\mid u\cdot v>0\\}$ is a positive cone on $A$. This bi-ordering
does not have any proper non-trivial convex subgroups. ∎

######  Example 2.16.

Suppose that $A$ is a free abelian group with a countably infinite basis
$\mathcal{B}$. Choose a total ordering of $\mathcal{B}$. For example, we could
index the elements of $\mathcal{B}$ by the integers or rational numbers. Every
element $x\in A$ can be represented uniquely as a linear sum of elements

$x=\sum_{v\in\mathcal{B}}k_{v}v$

where $\\{k_{v}\\}_{v\in\mathcal{B}}$ is a set of integer coefficients, all
but finite many of which are zero. For each $a$, let $v(a)$ be the maximal
basis element in $\mathcal{B}$ such that $a$ has a non-zero coefficient.
Define $P$ to be the set of all elements $a\in A$ such that the coefficient of
$v(a)$ is positive. ∎

We will now construct bi-orderings of a group by choosing a sequence of nested
normal subgroups in which every quotient of consecutive terms in a free
abelian group. Theorem 2.14 ensures that our quotients have bi-orders. We can
then use the bi-orders of the quotients to define a bi-ordering of the entire
group.

######  Construction 2.17.

Let $G$ be a group. Suppose that we have a nested sequence of normal subgroups
of $G$

$G=C_{1}\triangleright C_{2}\triangleright\cdots\triangleright C_{n}$

where $C_{i}/C_{i+1}$ is a free abelian group for $1\leq i<n$. For each $1\leq
i<n$, let $p_{i}\colon C_{i}\to C_{i}/C_{i+1}$ be the quotient map. Choose a
positive cone of $C_{n}$ invariant under conjugation by all elements of $G$.
For every $0\leq i<n$, choose a positive cone $P_{i}$ for the quotient
$C_{i}/C_{i+1}$ invariant under conjugation of elements of $G$ so for all
$g\in C_{i}$ and $x\in G$, we have $p_{i}(g)\in P_{i}$ if and only if
$p_{i}(xgx^{-1})\in P_{i}$. Define an element $g\in G$ to be positive when
either of the following occurs:

  1. (1)

$g\in C_{i}-C_{i+1}$ and $p_{i}(g)\in P_{i}$ for some $1\leq i<n$

  2. (2)

$g\in P_{n}$ ∎

We can extend this construction to an infinite nested sequence of subgroups
when their intersection is trivial.

######  Construction 2.18.

Let $G$ be a group. Suppose that we have a infinite nested sequence of normal
subgroups of $G$

$G=C_{1}\triangleright C_{2}\triangleright C_{3}\triangleright\cdots$

where $C_{i}/C_{i+1}$ is a free abelian group and $\bigcap C_{i}=\\{1\\}$. For
each $i$, let $p_{i}\colon C_{i}\to C_{i}/C_{i+1}$ be the quotient map. Choose
a positive cone $P_{i}$ for the quotient $C_{i}/C_{i+1}$ invariant under
conjugation by elements of $G$. Define an element $g\in G$ to be positive when
$g\in C_{i}-C_{i+1}$ and $p_{i}(g)\in P_{i}$ for some $i$. ∎

A key example of the use of this general strategy is in constructing bi-orders
on free groups.

######  Definition 2.19.

Let $G$ be a finite rank free group. Let $G_{1}=G$ and for $n>1$ let
$G_{n}=[G_{n-1},G]$. The sequence $\\{G_{n}\\}_{n\geq 1}$ is the _lower
central series_ of $G$. For $n\geq 1$, let $A_{n}=G_{n}/G_{n+1}$ be the _$n$
th lower central series quotient_. ∎

######  Remark 2.20.

Each $A_{n}$ is a free abelian group. Hall’s work [9] implies that the rank of
$A_{n}$ is less than or equal to $2^{n}$. ∎

######  Theorem 2.21 (Magnus [16]).

The set of terms of the lower central series of a free group has trivial
intersection. ∎

The following definition is due to Rolfsen [13, Appendix A].

######  Definition 2.22.

A _standard bi-order_ on a free group $G$ is one constructed as follows.
Applying Theorem 2.14, choose a positive cone $P_{n}$ on each lower central
series quotient $A_{n}$. By Theorem 2.21, for each nontrivial element $g\in G$
there is a unique positive integer $n(g)$ such that $g\in
G_{n(g)}-G_{n(g)+1}$. Let $[g]_{n}$ be the class of $g$ in $A_{n}$. We define
a positive cone $P$ on $G$ by:

(2.23) $P:=\\{g\in G\mid g\neq 1\text{ and }[g]_{n(g)}\in P_{n(g)}\\}.\qed$

######  Proposition 2.24.

A bi-ordering of the free group is standard if and only if every term of the
group’s lower central series is convex with respect to that bi-order.

###### Proof.

Let $(G,<,P)$ be a bi-ordered free group. If $P$ is standard, then by Lemma
2.5 the restriction of the bi-order to $G_{n-1}$ induces the bi-ordering of
$A_{n}$ given by $P_{n}$. Thus by Proposition 2.11, the next term $G_{n}$ is
convex in $G_{n-1}$. It follows by induction that $G_{n}$ is convex in $G$.

Suppose that $G_{n}$ is convex for all $n$. Then the bi-order of $G$ induces a
bi-order $<_{n}$ on $A_{n}:=G_{n}/G_{n+1}$ for each $n$. Define $P_{n}$ to be
the positive cone of $(A_{n},<_{n})$. $P$ can be described as the standard bi-
order of $G$ determined by the positive cones $P_{n}$. ∎

######  Proposition 2.25.

Suppose that $G$ is a finite rank free group and
$\phi,\phi^{\prime}\in\mathsf{Aut}(G)$ are automorphisms such that the induced
maps on the abelianization of $G$ are the same. Then, $\phi$ preserves a
standard bi-ordering of $G$ if and only if $\phi^{\prime}$ does.

###### Proof.

Consider a standard positive cone $P$ of $G$ determined by a collection of
positive cones $P_{n}$ of $A_{n}=G_{n}/G_{n+1}$. Given an automorphism
$\psi\in\mathsf{Aut}(G)$, denote by $\psi_{n}$ the induced automorphisms on
$A_{n}$. Then $\psi$ preserves $P$ if and only if $\psi_{n}(P_{n})=P_{n}$ for
all positive integers $n$. By a lemma of Perron and Rolfsen [17, Lemma 4.5],
there are canonical embeddings $\iota_{n}\colon A_{n}\hookrightarrow
A_{1}^{\otimes n}$ such that the following diagram commutes.

${A_{n}}$${A_{1}^{\otimes n}}$${A_{n}}$${A_{1}^{\otimes
n}}$$\scriptstyle{\iota_{n}}$$\scriptstyle{\psi_{n}}$$\scriptstyle{\psi_{1}^{\otimes
n}}$$\scriptstyle{\iota_{n}}$

Consider two automorphisms $\phi$ and $\phi^{\prime}$ of $G$ that induce the
same automorphism of the abelianization of $G$. In other words, we have that
$\phi_{1}=\phi_{1}^{\prime}$. Suppose that $\phi(P)=P$. Then for each $n$, we
have that $\phi_{n}(P_{n})=P_{n}$ Thus, it follows that $\phi_{1}^{\otimes
n}(\iota_{n}(P_{n}))=\iota_{n}(P_{n})$. Since $\phi_{1}$ is equal to
$\phi_{1}^{\prime}$, it follows that $\phi^{\prime}_{n}(P_{n})=P_{n}$ for each
$n$. Therefore, $\phi^{\prime}(P)=P$. ∎

##  3\. Punctured torus bundles

######  Definition 3.1.

We define the _(untwisted) punctured torus bundle_ $M$ over the circle with
monodromy $h$ as

$M\cong\frac{T\times[0,1]}{(x,1)\sim(h(x),0)}$

where $T$ is a torus with a boundary component and $h$ is an automorphism of
$T$ which fixes $\partial T$ pointwise. ∎

######  Proposition 3.2.

Suppose that $M$ is a hyperbolic punctured torus bundle. Then $\pi_{1}(M)$ is
bi-orderable if and only if $M$ is untwisted.

###### Proof.

For hyperbolic punctured torus bundles the Alexander polynomial has two real
roots. These roots have the same sign, which is positive in the untwisted case
and negative in the twisted case. For untwisted punctured torus bundles then,
$\pi_{1}(M)$ is bi-orderable by Theorem 1.6. Conversely, if $\pi_{1}(M)$ is
bi-orderable then by Theorem 1.7 at least one of the roots is positive, and
therefore the bundle is untwisted. ∎

Up to isotopy, the monodromy $h$ is a product of Dehn twists about two simple
closed curves on $T$ which intersect once. By choosing a base point $x$ on
$\partial T$, $h$ induces a well-defined automorphism $h_{*}$ of
$\pi_{1}(T,x)$ which is a rank 2 free group. The fundamental group of $M$ is
given by

$\pi_{1}(M)\cong G\rtimes_{h_{*}}\mathbb{Z}$

Here $G\cong\langle\alpha,\beta\rangle$ is the fundamental group of the
punctured torus, generated by loops $\alpha$ and $\beta$ as shown in Figure
3.1. Fixing notation, let $\mathbb{Z}\cong\langle\tau\rangle$.

2pt $\tau$ [r] at 80 142 $\alpha$ [br] at 154 65 $\beta$ [r] at 78 52 $h$ [l]
at 263 140

Figure 3.1. Generators of the fundamental group for a punctured torus bundle.

######  Remark 3.3.

Consider the figure eight knot complement, with monodromy given by a single
Dehn twist around a curve parallel to $\beta$ (as in Figure 3.1), followed by
a single Dehn twist around a curve parallel to $\alpha$. One can check that
the images of $\alpha$ and $\beta$ under the monodromy are
$\tau^{-1}\alpha\tau=h(\alpha)=\alpha\beta\alpha$ and
$\tau^{-1}\beta\tau=h(\beta)=\beta\alpha$. ∎

We will use the following well-known fact a few times. See Farb-Margalit [8,
Section 13.1] for details.

######  Lemma 3.4.

A pseudo-Anosov map $A\in\operatorname{SL}(2,\mathbb{Z})$ cannot have
eigenvalues of either $1$ or $-1$. ∎

######  Lemma 3.5.

$\mathsf{rk}\,H_{1}(M;\mathbb{Z})=1$

###### Proof.

Let $[\alpha]$ and $[\beta]$ be the integral homology classes of the curves
$\alpha$ and $\beta$, as in Figure 3.1. Let
$A\in\operatorname{SL}(2,\mathbb{Z})$ be the map $h_{+}$ on homology with
respect to the basis $[\alpha],[\beta]$. The homology group
$H_{1}(M;\mathbb{Z})$ decomposes as $H_{1}(M;\mathbb{Z})\cong
M_{T}\oplus\langle t\rangle$ where

$M_{T}=\langle[\alpha],[\beta]:[\alpha]=h_{+}[\alpha],[\beta]=h_{+}[\beta]\rangle.$

$M_{T}$ is torsion if and only if $\det(I-A)\neq 0$. Since $h$ is pseudo-
Anosov, 1 cannot be an eigenvalue of $A$ by Lemma 3.4. Therefore
$\det(I-A)=\mathsf{ch}_{A}(1)\neq 0$. Therefore, $M_{T}$ is torsion and
$\mathsf{rk}\,H_{1}(M;\mathbb{Z})=1$. ∎

##  4\. Maximal convex subgroups

###  4.1. Proof of Theorem 1.3

Recall that $G$ is the free group on two elements. The following proposition
tells us that any pair of bi-orders on a group $\pi_{1}(M)\cong
G\rtimes_{h}\mathbb{Z}$ either agree or precisely disagree at the level of the
$\mathbb{Z}$ term. (This is not the case for a general semidirect product of a
group with the integers.)

######  Proposition 4.1.

The maximal convex subgroup of any bi-ordering of $\pi_{1}(M)$ is $G$, the
fundamental group of the punctured torus.

###### Proof.

Let $<$ be a bi-order of $\pi_{1}(M)$. Since $\pi_{1}(M)$ is finitely
generated, it has a maximal convex subgroup $C$ by Proposition 2.12.

By Corollary 2.8, $H:=\pi_{1}(M)/C$ is a nontrivial bi-orderable subgroup. By
Theorem 2.13, $H$ is abelian. Thus, $H$ is isomorphic to a nontrivial bi-
orderable quotient of $H_{1}(M;\mathbb{Z})$. By Lemma 3.5, we have that
$\mathsf{rk}\,H_{1}(M;\mathbb{Z})=1$ and $H\cong\mathbb{Z}$. Thus
$\pi_{1}(M)/C\cong\mathbb{Z}$. Also $\pi_{1}(M)\cong G\rtimes_{h}\mathbb{Z}$,
so both $C$ and $G$ are kernels of surjections from $\pi_{1}(M)$ to
$\mathbb{Z}$. But because $\mathsf{rk}\,H_{1}(M;\mathbb{Z})=1$, there is a
unique (up to sign) surjection from $\pi_{1}(M)$ to $\mathbb{Z}$. Therefore
$C=G$. ∎

Thus a bi-ordering of $\pi_{1}(M)\cong G\rtimes_{h}\mathbb{Z}$ is determined
by a bi-ordering of the free group $G$ (together with a choice of which way we
order $\mathbb{Z}$). The next proposition tells us that when bi-ordering the
free group $G$ subject to invariance under $h$, we initially have no choice
but to follow the standard construction, as given in Definition 2.22.

######  Proposition 4.2.

Consider any bi-order $<$ on $\pi_{1}(M)$ and the induced bi-order $<_{G}$ on
$G$. The maximal convex subgroup of $G$ with respect to $<_{G}$ is
$G_{2}=[G,G]$.

###### Proof.

Let $<$ be a bi-order of $\pi_{1}(M)$. By Proposition 2.6, this induces a bi-
order $<_{G}$ on $G$ which is invariant under $h$. Since $G$ is finitely
generated, it has a maximal convex subgroup $C$ by Proposition 2.12.

By Theorem 2.13, $H:=G/C$ is a nontrivial abelian bi-orderable subgroup. Thus,
$H$ is a nontrivial bi-orderable quotient of $H_{1}(T;\mathbb{Z})$ so either
$H\cong\mathbb{Z}$ or $H\cong\mathbb{Z}^{2}$.

Assume for a contradiction that $H\cong\mathbb{Z}$. Let $p\colon G\to
H_{1}(T;\mathbb{Z})$ be the abelianization map. The monodromy $h$ induces a
map $h_{+}$ on $H_{1}(T;\mathbb{Z})$. By Proposition 2.12, $h(C)=C$ so
$h_{+}(p(C))=p(h(C))=p(C)$. Since $H\cong\mathbb{Z}$ then
$p(C)\cong\mathbb{Z}$, and since $h_{+}(p(C))=p(C)$ the eigenvalues are either
$1$ or $-1$. This is then a contradiction to Lemma 3.4. Therefore,
$H\cong\mathbb{Z}^{2}$. It follows that $C=G_{2}$. ∎

###### Proof of Theorem 1.3.

This follows from Propositions 4.1 and 4.2. ∎

###  4.2. Monodromy action on $G_{2}$

Consider the abelian cover $\pi\colon\hat{T}\to T$ corresponding to the
subgroup $G_{2}$. We orient the components of $\partial\hat{T}$
counterclockwise. Choose a component of $\partial\hat{T}$ and call it
$\gamma_{0}$. Let $\hat{x}$ be the lift of the basepoint $x$ on $\gamma_{0}$.
Elements of $G$ can be lifted to homotopy classes of paths in $\hat{T}$ based
at $\hat{x}$.

######  Definition 4.3.

Let $\hat{h}\colon\hat{T}\to\hat{T}$ be the lift of $h$ that preserves
$\gamma_{0}$. Let $\hat{h}_{+}$ be the map induced on
$H_{1}(\hat{T};\mathbb{Z})$ by $\hat{h}$. ∎

Figure 4.1 illustrates $\hat{h}$ in the example of the figure eight knot
complement.

2pt $\gamma_{0}$ [tr] at 150 150 $\hat{x}$ [bl] at 162 162

(a) The lift of the commutator $\alpha\beta\alpha^{-1}\beta^{-1}$ is homotopic
to a loop $\gamma_{0}$ in $\hat{T}$.

2pt $\gamma_{0}$ [tr] at 222 150 $\hat{x}$ [bl] at 234 162

(b) The image $\hat{h}(\gamma_{0})$ is homotopic to $\gamma_{0}$.

Figure 4.1. The map $\hat{h}$ acts on the abelian cover $\hat{T}$, and
preserves $\gamma_{0}$. Here we show the image for the figure eight knot
complement. By Remark 3.3, $h(\alpha)=\alpha\beta\alpha$ and
$h(\beta)=\beta\alpha$. Note that $\gamma_{0}$ is fixed by $\hat{h}$.

######  Definition 4.4.

Let $\mathcal{B}$ be the countable basis of $H_{1}(\hat{T},\mathbb{Z})$
consisting of the homology classes corresponding to the components of
$\partial\hat{T}$. ∎

######  Notation 4.5.

To lighten the notation, we will use the symbol $\gamma$ for a generic
component of $\partial\hat{T}$ and also for its homology class. ∎

Recall that $\pi$ is the covering map from $\hat{T}$ to $T$.

######  Lemma 4.6.

Suppose that $\gamma$ is a component of $\partial\hat{T}$. Let $\eta$ be a
path in $\hat{T}$ from $\gamma_{0}$ to $\gamma$. Then the class
$[\pi(\eta)]\in H_{1}(T;\mathbb{Z})$ is independent of the choice of $\eta$.

###### Proof.

Suppose that $\eta$ and $\eta^{\prime}$ are two such paths. Then the
concatenation $\eta^{\prime}\cdot\eta^{-1}$ is a loop in $\hat{T}$. Thus
$\pi(\eta^{\prime}\cdot\eta^{-1})\in[G,G]$. Therefore
$0=[\pi(\eta^{\prime}\cdot\eta^{-1})]=[\pi(\eta^{\prime})\cdot\pi(\eta^{-1})]=[\pi(\eta^{\prime})\cdot\pi(\eta)^{-1}]=[\pi(\eta^{\prime})]-[\pi(\eta)]$.
Thus $[\pi(\eta)]=[\pi(\eta^{\prime})]$. ∎

We may therefore make the following definition.

######  Definition 4.7.

For any choice of $\eta$ as in Lemma 4.6, we define

$q:\mathcal{B}\to
H_{1}(T;\mathbb{Z})\cong\mathbb{Z}^{2}\subset\mathbb{R}^{2}\quad\mbox{by}\quad
q(\gamma)=[\pi(\eta)].\qed$

######  Lemma 4.8.

The map $q$ is a bijection.

###### Proof.

Recall that the curves $\alpha$ and $\beta$ are our generators for $G$, the
fundamental group of the punctured torus $T$, as shown in Figure 3.1. For any
element $m[\alpha]+n[\beta]$ of $H_{1}(T;\mathbb{Z})$, if we lift the loop
$\alpha^{m}\cdot\beta^{n}$ to $\hat{T}$ we find a path based at $\hat{x}$
which ends on a loop $\gamma$ such that $q(\gamma)=m[\alpha]+n[\beta]$. Thus
$q$ is surjective.

Now suppose that $\gamma$ and $\gamma^{\prime}$ are elements of
$\partial\hat{T}$ with $q(\gamma)=q(\gamma^{\prime})$. Let $\eta$ and
$\eta^{\prime}$ be paths for $\gamma$ and $\gamma^{\prime}$ as in Lemma 4.6.
So $[\pi(\eta)]=[\pi(\eta^{\prime})]$. Following the calculation in the proof
of Lemma 4.6 backwards, we deduce that
$\pi(\eta^{\prime}\cdot\eta^{-1})\in[G,G]$. This implies that
$\eta^{\prime}\cdot\eta^{-1}$ is a loop in $\hat{T}$, and so
$\gamma=\gamma^{\prime}$. Thus $q$ is injective. ∎

######  Lemma 4.9.

We have the following.

  1. (1)

$\hat{h}_{+}(\mathcal{B})\subset\mathcal{B}$

  2. (2)

$q\circ\hat{h}_{+}=h_{+}\circ q$

###### Proof.

Since $\hat{h}$ is a self-homeomorphism of $\hat{T}$, it sends boundary loops
to boundary loops. Thus $\gamma\in\mathcal{B}$ implies that
$\hat{h}_{+}(\gamma)\in\mathcal{B}$ and we have (1).

Recall that $\pi\colon\hat{T}\to T$ is the covering map. Then by definition,
$h\circ\pi=\pi\circ\hat{h}$. Let $\gamma\in\mathcal{B}$ and let $\eta$ be a
path in $\hat{T}$ from $\gamma_{0}$ to $\gamma$. Then

$h_{+}\circ
q(\gamma)=h_{+}([\pi(\eta)])=[h\circ\pi(\eta)]=[\pi(\hat{h}(\eta))]$

Note that $\hat{h}(\eta)$ is a path based at $\hat{x}$ which ends on
$\hat{h}(\gamma)$. So we get that
$[\pi(\hat{h}(\eta))]=q([\hat{h}(\gamma)])=q\circ\hat{h}_{+}[(\gamma)]$ and we
have proved (2). ∎

######  Definition 4.10.

Let $\gamma\in\mathcal{B}$. The _shift_ of $\gamma$ by $(m,n)$ is

$s_{(m,n)}(\gamma)=q^{-1}\big{(}(m,n)+q(\gamma)\big{)}.\qed$

######  Definition 4.11.

Let $p_{1}:G\to\mathbb{Z}^{2}$ be the abelianization map. Let $p_{2}:G_{2}\to
G_{2}/[G_{2},G_{2}]$ be the quotient map. ∎

######  Lemma 4.12.

Suppose that $g\in G_{2}$ and $x\in G$. Suppose that the homology class
$p_{1}(x)$ in $H_{1}(T,\mathbb{Z})$ is $m[\alpha]+n[\beta]$ and that the
homology class $p_{2}(g)$ in $H_{1}(\hat{T},\mathbb{Z})$ is

$\sum_{\gamma\in\mathcal{B}}k_{\gamma}\gamma.$

Then the homology class of $p_{2}(xgx^{-1})$ in $H_{1}(\hat{T},\mathbb{Z})$ is

$\sum_{\gamma\in\mathcal{B}}k_{\gamma}s_{(m,n)}(\gamma)$

where $s_{(m,n)}(\gamma)$ is the shift of $\gamma$ by $(m,n)$, as in
Definition 4.10.

###### Proof.

First note that the set $\\{y\alpha\beta\alpha^{-1}\beta^{-1}y^{-1}\mid y\in
G\\}$ generates $G_{2}$. Therefore we can write

$g=\prod_{i}y_{i}(\alpha\beta\alpha^{-1}\beta^{-1})^{\pm 1}y_{i}^{-1}$

for some finite collection of $y_{i}\in G$. For any $y\in G$, we have that the
element $\gamma=p_{2}(y\alpha\beta\alpha^{-1}\beta^{-1}y^{-1})$ is a member of
$\mathcal{B}$. Thus we can write

$p_{2}(g)=\sum_{i}p_{2}(y_{i}(\alpha\beta\alpha^{-1}\beta^{-1})^{\pm
1}y_{i}^{-1})=\sum_{\gamma\in\mathcal{B}}k_{\gamma}\gamma$

where $k_{\gamma}$ is found by summing the $\pm 1$ exponents of the terms in
which the path $y_{i}$ connects $\gamma_{0}$ to $\gamma$ in $\hat{T}$.
Conjugating by $x$, we can write

$xgx^{-1}=\prod_{i}xy_{i}(\alpha\beta\alpha^{-1}\beta^{-1})^{\pm
1}y_{i}^{-1}x^{-1}$

Recall that the homology class $p_{1}(x)$ is $m[\alpha]+n[\beta]\in
H_{1}(T,\mathbb{Z})$. It follows that if
$p_{2}(y\alpha\beta\alpha^{-1}\beta^{-1}y^{-1})=\gamma$ then
$p_{2}(xy\alpha\beta\alpha^{-1}\beta^{-1}y^{-1}x^{-1})$ is the shift
$s_{(m,n)}(\gamma)$. Thus

$p_{2}(xgx^{-1})=\sum_{i}p_{2}(xy_{i}(\alpha\beta\alpha^{-1}\beta^{-1})^{\pm
1}y_{i}^{-1}x^{-1})=\sum_{\gamma\in\mathcal{B}}k_{\gamma}s_{(m,n)}(\gamma).\qed$

###  4.3. Proof of Theorem 1.4

We will prove Theorem 1.4 through a series of lemmas. First note that since
$C$ is convex, $<_{2}$ induces a bi-ordering of $G_{2}/C$. By Theorem 2.13,
there is an injective homomorphism $f\colon G_{2}/C\to\mathbb{R}$ respecting
the bi-orders. Also, since $G_{2}/C$ is abelian the quotient map factors
through quotient maps $p_{2}\colon G_{2}\to H_{1}(\hat{T};\mathbb{Z})\cong
G_{2}/[G_{2},G_{2}]$ and $\psi\colon H_{1}(\hat{T};\mathbb{Z})\to G_{2}/C$.
Define $\mu\colon H_{1}(\hat{T};\mathbb{Z})\to\mathbb{R}$ to be $f\circ\psi$
as in the following diagram.

${G_{2}}$${H_{1}(\hat{T};\mathbb{Z})}$${G_{2}/C}$${\mathbb{R}}$$\scriptstyle{p_{2}}$$\scriptstyle{\psi}$$\scriptstyle{\mu}$$\scriptstyle{f}$

Let $P$ be the positive cone associated to $<_{2}$. Let
$Q^{+}=\mu^{-1}(\mathbb{R}^{+})$ and $Q^{-}=\mu^{-1}(\mathbb{R}^{-})$. Since
$\mu\circ p_{2}$ and $<_{2}$ induce the bi-ordering of the image of $\mu$, we
have

$Q^{+}=p_{2}(P-C),\quad
Q^{-}=p_{2}(P^{-1}-C),\quad\mbox{and}\quad\ker\mu=p_{2}(C).$

######  Lemma 4.13.

Suppose that $\phi\in\mathsf{Aut}(G_{2})$ preserves $C$ and the bi-ordering
$(<_{2},P)$. Let $\phi_{+}\in\mathsf{Aut}(H_{1}(\hat{T};\mathbb{Z}))$ be the
automorphism induced by $\phi$. Then the sets $Q^{+}$, $Q^{-}$, and $\ker\mu$
are invariant under $\phi_{+}$.

###### Proof.

By assumption $\phi$ preserves $P$ and $C$. Therefore, by the definition of an
induced map, $\phi_{+}$ preserves $p_{2}(P-C)$, $p_{2}(P^{-1}-C)$, and
$p_{2}(C)$. ∎

In other words, for all $v\in H_{1}(\hat{T};\mathbb{Z})$, the sign of $\mu(v)$
is the same as the sign of $\mu(\phi_{+}(v))$.

######  Lemma 4.14.

Let $v,w\in H_{1}(\hat{T};\mathbb{Z})$ with $\mu(v)\neq 0$ and $\mu(w)\neq 0$.
Suppose that $\phi\in\mathsf{Aut}(G_{2})$ preserves $C$ and the bi-ordering
$<_{2}$. Let $\phi_{+}\in\mathsf{Aut}(H_{1}(\hat{T};\mathbb{Z}))$ be the
automorphism induced by $\phi$. Then

$\frac{\mu(v)}{\mu(w)}=\frac{\mu(\phi_{+}(v))}{\mu(\phi_{+}(w))}.$

###### Proof.

Let $d=\mu(v)/\mu(w)$, and define the set $\mathbb{Q}_{>d}$ as follows.

$\mathbb{Q}_{>d}=\\{r\in\mathbb{Q}\mid r>d\\}$

Thus, $d=\inf\mathbb{Q}_{>d}$. Consider an arbitrary element of
$\mathbb{Q}_{>d}$. Write this element as $a/b$ where $a,b$ are integers with
$b>0$. Thus $d<a/b$. We first assume that $\mu(w)>0$. Then we have that

$0=d\mu(w)-\mu(v)<(a/b)\mu(w)-\mu(v)=a\mu(w)-b\mu(v)$ $=\mu\big{(}aw-
bv\big{)}.$

By Lemma 4.13,

$0<\mu\big{(}\phi_{+}(aw)-bv)\big{)}=a\mu\big{(}\phi_{+}(w)\big{)}-b\mu\big{(}\phi_{+}(v)\big{)}$

Thus,

$\frac{\mu(\phi_{+}(v))}{\mu(\phi_{+}(w))}<\frac{a}{b}$

Since $a/b$ was arbitrary, $\mu(\phi_{+}(v))/\mu(\phi_{+}(w))$ is a lower
bound for $\mathbb{Q}_{>d}$. Therefore,

$\frac{\mu(\phi_{+}(v))}{\mu(\phi_{+}(w))}\leq d$

Similarly, $\mu(\phi_{+}(v))/\mu(\phi_{+}(w))$ is an upper bound for

$\mathbb{Q}_{<d}=\\{r\in\mathbb{Q}\mid r<d\\}$

so

$\frac{\mu(\phi_{+}(v))}{\mu(\phi_{+}(w))}=d=\frac{\mu(v)}{\mu(w)}.$

A similar argument goes through when $\mu(w)<0$. ∎

The map $\mu$ is determined by its value on the basis $\mathcal{B}$. For each
pair of integers $m,n$, define $\gamma_{m,n}$ to be $q^{-1}(m,n)$, where $q$
is as given in Definition 4.7. Thus,

$\gamma_{m,n}=s_{m,n}(\gamma_{0,0})$

######  Lemma 4.15.

For all $m,n\in\mathbb{Z}$,

$\mu(\gamma_{m,n})=\mu(\gamma_{0,0})a^{m}b^{n}$

for some $a,b\in\mathbb{R}^{+}$.

###### Proof.

For each pair of integers $m,n$,

$\gamma_{m,n}=p_{2}(\alpha^{m}\beta^{n}[\alpha,\beta]\beta^{-n}\alpha^{-m}]).$

Since conjugation preserves every positive cone of the free group $G$, Lemma
4.13 implies that either $\mu(\mathcal{B})\subset\mathbb{R}^{+}$,
$\mu(\mathcal{B})\subset\mathbb{R}^{-}$, or $\mu(\mathcal{B})=\\{0\\}$. Since
$\mu$ is not a trivial map, $\mu(\gamma_{m,n})\neq 0$ for all integer pairs
$m$ and $n$.

Let $a$ and $b$ be defined as follows.

$a=\frac{\mu(\gamma_{1,0})}{\mu(\gamma_{0,0})}\quad\mbox{and}\quad
b=\frac{\mu(\gamma_{0,1})}{\mu(\gamma_{0,0})}$

By Lemma 4.14,

$a=\frac{\mu(\gamma_{m+1,n})}{\mu(\gamma_{m,n})}\quad\mbox{and}\quad
b=\frac{\mu(\gamma_{m,n+1})}{\mu(\gamma_{m,n})}$

for all $m$ and $n$. It follows that
$\mu(\gamma_{m,n})=\mu(\gamma_{0,0})a^{m}b^{n}$. ∎

The following lemma is proved using _basic commutators_ , see [6, Chapter 3]
for the details.

######  Lemma 4.16.

$G_{2}/G_{3}\cong\mathbb{Z}$.

###### Proof.

By Theorem 3.1 of Clement-Majewics-Zyman [6], any element of $G_{2}/G_{3}$ is
generated modulo $G_{3}$ by the basic commutators of weight two. In our case
there is only one commutator of weight two, namely $[\beta,\alpha]$. The
result follows. ∎

######  Lemma 4.17.

If $\mu$ is constant on $\mathcal{B}$, then $C=G_{3}$.

###### Proof.

Suppose that $\mu$ is constant on $\mathcal{B}$. First we show that
$G_{3}\subset C$. By the first isomorphism theorem, $C=\ker(\phi\circ
p_{2})=\ker\mu\circ p_{2}$. It is sufficient to show that for arbitrary
elements $g\in G_{2}$ and $x\in G$, the element $gxg^{-1}x^{-1}\in G_{3}$ is
in $\ker\mu\circ p_{2}$. To show this, suppose that the homology class
$p_{1}(x)$ in $H_{1}(T,\mathbb{Z})$ is $m[\alpha]+n[\beta]$ and that the
homology class $p_{2}(g)$ in $H_{1}(\hat{T},\mathbb{Z})$ is

$\sum_{\gamma\in\mathcal{B}}k_{\gamma}\gamma.$

By Lemma 4.12,

$p_{2}(gxg^{-1}x^{-1})=\sum_{\gamma\in\mathcal{B}}k_{\gamma}\gamma-\sum_{\gamma\in\mathcal{B}}k_{\gamma}s_{(m,n)}(\gamma)=\sum_{\gamma\in\mathcal{B}}k_{\gamma}(\gamma-
s_{(m,n)}(\gamma)).$

Since $\mu$ is constant on $\mathcal{B}$, we have that

$\mu(p_{2}(gxg^{-1}x^{-1}))=\sum_{\gamma\in\mathcal{B}}k_{\gamma}\big{[}\mu(\gamma)-\mu(s_{(m,n)}(\gamma))\big{]}=0.$

Thus, $G_{3}\subset C$. By the third isomorphism theorem,

$\frac{G_{2}/G_{3}}{C/G_{3}}\cong\frac{G_{2}}{C}.$

By Lemma 4.16, we have that $G_{2}/C$ is a non-trivial bi-orderable quotient
of $\mathbb{Z}$. Since $\mathbb{Z}$ is the only bi-orderable cyclic group,
$C/G_{3}$ must be trivial. Therefore, $C=G_{3}$. ∎

Recall that from Definition 4.3, we lift the monodromy $h$ to a map $\hat{h}$
on the cover $\hat{T}$. Then $\hat{h}_{+}$ is the induced map on homology.

######  Lemma 4.18.

For all $v\in H_{1}(\hat{T};\mathbb{Z})$,

$\mu(\hat{h}_{+}(v))=\mu(v).$

###### Proof.

Let $v\in H_{1}(\hat{T};\mathbb{Z})$. The map $\hat{h}_{+}$ is induced by
$h_{*}$, and by assumption, $h_{*}$ preserves $P$. By Lemma 4.13, $\mu(v)=0$
if and only if $\mu(\hat{h}_{+}(v))=0$.

When $\mu(v)\neq 0$,

(4.19)
$\frac{\mu(v)}{\mu(\gamma_{0,0})}=\frac{\mu(\hat{h}_{+}(v))}{\mu(\hat{h}_{+}(\gamma_{0,0}))}$

by Lemma 4.14. Since $\gamma_{0,0}$ is fixed by $\hat{h}_{+}$, we have
$\mu(\gamma_{0,0})=\mu(\hat{h}_{+}(\gamma_{0,0}))$. Therefore, it follows from
(4.19) that $\mu(\hat{h}_{+}(v))=\mu(v)$. ∎

######  Lemma 4.20.

Suppose for $i=1,2,3$ that $\gamma_{m_{i},n_{i}}\in\mathcal{B}$ with

$\mu(\gamma_{m_{1},n_{1}})=\mu(\gamma_{m_{2},n_{2}})=\mu(\gamma_{m_{3},n_{3}}).$

If the points $\\{(m_{i},n_{i})\mid i=1,2,3\\}$ are not collinear in
$\mathbb{Z}^{2}$ then $\mu$ is constant on $\mathcal{B}$.

###### Proof.

By Lemma 4.15, there are numbers $a,b\in\mathbb{R}^{+}$ such that

$\mu(\gamma_{m,n})=\mu(\gamma_{0,0})a^{m}b^{n}$

for all integer pairs $m$ and $n$. Then

$\log\circ\mu(\gamma_{m,n})=\log\circ\mu(\gamma_{0,0})+m\log(a)+n\log(b)$

Thus $\log\circ\mu$ is a linear function of $m$ and $n$. Therefore if it is
constant at three non-collinear points then it is constant. ∎

###### Proof of Theorem 1.4.

Suppose that $M$ hyperbolic so $h$ is a pseudo-Anosov map. Consider the action
$\hat{h}_{+}$ on $\mathcal{B}$. The induced map $h_{+}$ doesn’t preserve any
proper non-trivial subgroups of $H_{1}(T;\mathbb{Z})=\mathbb{Z}^{2}$. Thus,
none of the orbits of $\mathcal{B}$ under $\hat{h}_{+}$ contain all collinear
points unless the orbit is a single fixed point.

Consider $\gamma_{m,n}\in\mathcal{B}-\\{\gamma_{0,0}\\}$. Since $\gamma_{m,n}$
is not a fixed point of $\hat{h}_{+}$, the orbit of $\gamma_{m,n}$ contains
three distinct elements $\gamma$, $\gamma^{\prime}$, and
$\gamma^{\prime\prime}$ which are not collinear. By Lemma 4.18,

$\mu(\gamma)=\mu(\gamma^{\prime})=\mu(\gamma^{\prime\prime}).$

Therefore, by Lemma 4.20 the function $\mu$ is constant on $\mathcal{B}$, and
by Lemma 4.17 we have that $C=G_{3}$. ∎

##  5\. A non-standard bi-order

To prove Theorem 1.5, we construct non-standard bi-orders on untwisted
hyperbolic punctured torus bundles following Construction 2.17.

###  5.1. The first quotient

The map $h\colon T\to T$ induces a map
$h_{+}\in\operatorname{SL}(2,\mathbb{Z})$ on
$H_{1}(T;\mathbb{Z})\cong\mathbb{Z}^{2}$ with eigenvalues
$(\lambda,1/\lambda)$ and corresponding eigenbasis
$(e_{\lambda},e_{1/\lambda})$. The bundle is untwisted, so the eigenvalues are
positive. Using this eigenbasis, we can write $h_{+}$ as the matrix

$\mathopen{}\mathclose{{}\left(\begin{array}[]{cc}\lambda&0\\\
0&1/\lambda\end{array}}\right).$

It follows that $h_{+}$ preserves four positive cones $Q_{1}$ of
$\mathbb{Z}^{2}$. Each of these is given by taking one of
$\\{e_{\lambda},e_{1/\lambda},-e_{\lambda},-e_{1/\lambda}\\}$ for the vector
$v$ in Example 2.15(2). See Figure 5.1.

(a) $e_{\lambda}$ (b) $-e_{\lambda}$ (c) $e_{1/\lambda}$ (d) $-e_{1/\lambda}$

Figure 5.1. The four positive cones (shaded) in the example of the figure
eight knot complement.

###  5.2. The second quotient

Recall the definition of the map $\hat{h}$ and basis of $\mathcal{B}$ from
Definitions 4.3 and 4.4. The goal is to define a positive cone of
$H_{1}(\hat{T};\mathbb{Z})$ invariant under $\hat{h}_{+}$. In short, the plan
is to find an ordering on $\mathcal{B}$ that is invariant under $\hat{h}_{+}$,
and then define a bi-ordering of $H_{1}(\hat{T};\mathbb{Z})$ in a
lexicographic fashion relative to the ordering of $\mathcal{B}$ as in Example
2.16.

######  Definition 5.1.

Let $e$ be one of
$\\{e_{\lambda},e_{1/\lambda},-e_{\lambda},-e_{1/\lambda}\\}$. Again using
Example 2.15(2), we generate an order $\prec_{e}$ on $\mathbb{Z}^{2}$
(possibly different from the choice made in Section 5.1). This order induces
an ordering $<_{e}$ on $\mathcal{B}$ via $q$. That is,
$\gamma_{1}<_{e}\gamma_{2}$ if and only if
$q(\gamma_{1})\prec_{e}q(\gamma_{2})$. ∎

######  Lemma 5.2.

The order $<_{e}$ on $\mathcal{B}$ is invariant under shifting by any vector
$(m,n)\in\mathbb{Z}^{2}$.

###### Proof.

Addition preserves the order $\prec_{e}$ on $\mathbb{Z}^{2}$. The result then
follows from Definition 4.10. ∎

######  Lemma 5.3.

The order $<_{e}$ on $\mathcal{B}$ is invariant under $\hat{h}_{+}$.

###### Proof.

Let $\gamma_{1}$ and $\gamma_{2}$ be elements of $\mathcal{B}$. Suppose that
$\gamma_{1}<_{e}\gamma_{2}$. Therefore $q(\gamma_{1})\prec_{e}q(\gamma_{2})$.
Since $\prec_{e}$ is invariant under the action of $h_{+}$ we have that

$h_{+}\circ q(\gamma_{1})\prec_{e}h_{+}\circ q(\gamma_{2}).$

By Lemma 4.9(2), we have

$q\circ\hat{h}_{+}(\gamma_{1})\prec_{e}q\circ\hat{h}_{+}(\gamma_{2}),$

and so

$\hat{h}_{+}(\gamma_{1})<_{e}\hat{h}_{+}(\gamma_{2}).\qed$

We next define a positive cone $Q_{2}$ on $H_{1}(\hat{T};\mathbb{Z})$ as
follows.

######  Definition 5.4.

Given a non-zero element $v\in H_{1}(\hat{T};\mathbb{Z})$, we define $\max(v)$
to be the maximum (under $<_{e}$) basis element $\gamma\in\mathcal{B}$ such
that the coefficient of $\gamma$ in $v$ is nonzero. Let $k(v)$ be this
coefficient. We then define

$Q_{2}=\\{v\in H_{1}(\hat{T};\mathbb{Z})\mid k(v)>0\\}.\qed$

######  Lemma 5.5.

The set $Q_{2}$ is a positive cone. Moreover $Q_{2}$ is invariant under the
action of $\hat{h}_{+}$.

###### Proof.

We check the three properties in Definition 2.1. First suppose that $v,w\in
Q_{2}$. If $\max(v)=\max(w)$ then $\max(v+w)=\max(v)=\max(w)$ and
$k(v+w)=k(v)+k(w)>0$. If not, then breaking symmetry, assume that
$\max(v)<_{e}\max(w)$. In this case, $\max(v+w)=\max(w)$ and $k(v+w)=k(w)>0$.
This proves (1). Now suppose that $v\in H_{1}(\hat{T};\mathbb{Z})-\\{0\\}$.
Note that $\max(-v)=\max(v)$ and $k(-v)=-k(v)$. Therefore $v\in Q_{2}$ if and
only if $-v\in Q_{2}^{-1}$. By definition, $0\notin Q_{2}\cup Q_{2}^{-1}$, so
we have (2). Since $H_{1}(\hat{T};\mathbb{Z})$ is abelian, we have (3).

Finally we show that $Q_{2}$ is invariant under $\hat{h}_{+}$. Let $v\in
Q_{2}$. Thus $k(v)>0$. For some set of coefficients
$\\{k_{\gamma}\\}_{\gamma\in\mathcal{B}}$,

$v=\sum_{\gamma\in\mathcal{B}}k_{\gamma}\gamma.$

Thus,

$\hat{h}_{+}(v)=\sum_{\gamma\in\mathcal{B}}k_{\gamma}\hat{h}_{+}(\gamma).$

By Lemma 5.3, the ordering of elements in $\mathcal{B}$ is invariant under
$\hat{h}_{+}$. Therefore, $\max(v)=\max(\hat{h}_{+}(v))$. Thus
$k(\hat{h}_{+}(v))=k(v)>0$. Therefore $\hat{h}_{+}(v)\in Q_{2}$. ∎

By the definition of $\hat{T}$, the fundamental group $\pi_{1}(\hat{T})$ is
isomorphic to $G_{2}$. Abelianizing, we see that $G_{2}/[G_{2},G_{2}]$ is
naturally isomorphic to $H_{1}(\hat{T},\mathbb{Z})$. Under this isomorphism,
$Q_{2}$ defines a positive cone of $G_{2}/[G_{2},G_{2}]$.

###  5.3. The remainder

We are now ready to build a bi-ordering of the free group $G$ that is
invariant under the action of $h$. We require one more piece of data: we
choose an arbitrary positive cone of $\pi_{1}(M)$ (recall that in the
statement of Theorem 1.5 we assume that $\pi_{1}(M)$ is bi-orderable). We
refer to the restriction of this arbitrary positive cone to $[G_{2},G_{2}]$ as
$Q_{3}$.

###  5.4. Putting it all together

We now apply Construction 2.17 to build a bi-ordering of the free group $G$,
invariant under the action of $h$, as follows. Recall that
$p_{1}:G\to\mathbb{Z}^{2}$ is the abelianization map and that $p_{2}:G_{2}\to
G_{2}/[G_{2},G_{2}]$ is the quotient map.

######  Definition 5.6.

Define $P$ to be the set of all $g\in G$ such that either

  1. (1)

$p_{1}(g)\in Q_{1}$, or

  2. (2)

$g\in G_{2}$ and $p_{2}(g)\in Q_{2}$, or

  3. (3)

$g\in Q_{3}$. ∎

######  Lemma 5.7.

$P$ is a positive cone of $G$.

###### Proof.

The goal is to show that $P$ is a positive cone by repeatedly applying Lemma
2.5. By definition, $Q_{3}$ is a positive cone of $[G_{2},G_{2}]$ and is
invariant under conjugation. We first apply Lemma 2.5 with $p_{2}$ as the
surjective homomorphism, mapping from $G_{2}$ to $G_{2}/[G_{2},G_{2}]$. We
obtain that $P\cap G_{2}$ is a positive cone on $G_{2}$.

Next, we want to apply Lemma 2.5 with $p_{1}$ as the surjective homomorphism,
mapping from $G$ to $G/G_{2}\cong H_{1}(T;\mathbb{Z})\cong\mathbb{Z}^{2}$. In
order to do so, we need to show that $P\cap G_{2}$ is invariant under
conjugation by the elements of $G$.

Suppose that $g\in G_{2}\cap P$ and that $x\in G$. There are two cases. If
$g\in Q_{3}$ then again by definition $xgx^{-1}\in Q_{3}\subset G_{2}\cap P$.
Otherwise we have that $g\in G_{2}$ and $p_{2}(g)\in Q_{2}$. In this case,
suppose that the homology class $p_{1}(x)$ is $m[\alpha]+n[\beta]\in
H_{1}(T,\mathbb{Z})$, and the homology class $p_{2}(g)$ in
$H_{1}(\hat{T},\mathbb{Z})$ is

$\sum_{\gamma\in\mathcal{B}}k_{\gamma}\gamma.$

By Lemma 4.12, the homology class of $p_{2}(xgx^{-1})$ in
$H_{1}(\hat{T},\mathbb{Z})$ is

$\sum_{\gamma\in\mathcal{B}}k_{\gamma}s_{(m,n)}(\gamma).$

Lemma 5.2 implies that the maximal element of $\mathcal{B}$ appearing in the
sum for $x^{-1}gx$ is the shift of the maximal element appearing in the sum
for $g$. The coefficients for these elements are equal, and therefore have the
same sign. By Definition 5.4, we have that $p_{2}(g)\in Q_{2}$ if and only if
$p_{2}(x^{-1}gx)\in Q_{2}$. Thus we have obtained the hypotheses of Lemma 2.5
and we are done. ∎

######  Lemma 5.8.

The positive cone $P$ is invariant under the map induced by the monodromy
$h_{*}$.

###### Proof.

Suppose that $g\in P$. There are three possibilities, as given in Definition
5.6. First suppose that $p_{1}(g)\in Q_{1}$. Since $p_{1}\circ
h_{*}(g)=h_{+}\circ p_{1}(g)$ and by definition, $h_{+}$ preserves $Q_{1}$, we
have that $p_{1}(h_{*}(g))\in Q_{1}$.

Next suppose that $g\in G_{2}$ and $p_{2}(g)\in Q_{2}$. Since $G_{2}$ is
characteristic, $h_{*}(g)\in G_{2}$. The positive cone $Q_{2}$ is invariant
under $\hat{h}_{*}$ by Lemma 5.5, so $\hat{h}_{*}\circ p_{2}(g)\in Q_{2}$.
Therefore $p_{2}\circ h_{*}(g)=\hat{h}_{*}\circ p_{2}(g)\in Q_{2}$.

Finally, suppose that $g\in Q_{3}$. Since $Q_{3}$ is the restriction of a
positive cone of $\pi_{1}(M)$ and $[G_{2},G_{2}]$ is characteristic, $Q_{3}$
is invariant under $h_{*}$. Thus, $h_{*}(g)\in Q_{3}$. ∎

######  Lemma 5.9.

$G_{3}=[G_{2},G]$ in the lower central series for $G$ is not convex with
respect to $P$.

###### Proof.

We will apply Proposition 2.7, finding elements $y\in P$ and $z\in P^{-1}$ in
the same nontrivial coset of $G_{3}$.

Let $\gamma_{0}=\alpha\beta\alpha^{-1}\beta^{-1}$, and let
$\gamma_{1}=\beta\alpha^{-1}\beta^{-1}\alpha\in G_{2}=[G,G]$. Since
$\gamma_{0}=[\alpha,\gamma_{1}]\gamma_{1}$, we have that $\gamma_{0}$ and
$\gamma_{1}$ are in the same coset $S=\gamma_{0}G_{3}=\gamma_{1}G_{3}$.
(Recall that $G_{3}$ is normal, so left and right cosets are the same.)
Consider the elements $y=\gamma_{0}^{2}\gamma_{1}^{-1}$ and
$z=\gamma_{0}^{-1}\gamma_{1}^{2}$. Since $\gamma_{0}$ and $\gamma_{1}$ are in
$S$, we have that $\gamma_{0}\gamma_{1}^{-1}$ and $\gamma_{0}^{-1}\gamma_{1}$
are in $G_{3}$. Thus $y$ and $z$ are also in $S$.

By the definition of $\hat{T}$, we have that
$G_{2}\cong\pi_{1}(\hat{T},\hat{x})$. We define the _winding number_ of an
element $f\in G_{2}$ by

$w(f)=\sum_{\gamma\in\mathcal{B}}k_{\gamma}$

where

$p_{2}(f)=\sum_{\gamma\in\mathcal{B}}k_{\gamma}\gamma.$

The group $G_{3}$ is generated by terms $[g,f]$ where $g\in G$ and $f\in
G_{2}$. By Lemma 4.12, we have that $w(gfg^{-1})=w(f)$, so $w([g,f])=0$.
Therefore $G_{3}$ is contained in $\ker(w)$. Since
$w(\gamma_{0})=w(\gamma_{1})=1$, we have that $S\neq G_{3}$.

The homology class $p_{2}(y)=2p_{2}(\gamma_{0})-p_{2}(\gamma_{1})$, and the
homology class $p_{2}(z)=-p_{2}(\gamma_{0})+2p_{2}(\gamma_{1})$. Since
$p_{2}(\gamma_{0})$ and $p_{2}(\gamma_{1})$ are in $\mathcal{B}$ and the signs
of the coefficients of $p_{2}(y)$ and $p_{2}(z)$ are opposite, exactly one of
the homology classes is in $Q_{2}$ and the other is in $Q_{2}^{-1}$.
Therefore, one of $y$ and $z$ is in $P$ and the other is in $P^{-1}$. ∎

###### Proof of Theorem 1.5.

Lemma 5.7 tells us that $P$ is a positive cone. From Lemma 5.9 and Proposition
2.24 we deduce that $P$ is not the positive cone of a standard bi-order of
$G$. ∎

######  Corollary 5.10.

Let $P$ be the positive cone given in Definition 5.6. Then $G_{2}$ has no
maximal convex subgroup with respect to $P$.

###### Proof.

This follows from Lemma 5.9 and Theorem 1.4. It can also be seen directly by
exhibiting an increasing sequence of convex subgroups. Namely, for each
$\gamma\in\mathcal{B}$, let

$C_{\gamma}=\\{g\in G_{2}\mid\max(p_{2}(g))=\gamma\\}\cup\\{\text{id}_{G}\\}$

where $m$ is the function given in Definition 5.4. Each $C_{\gamma}$ is a
convex subgroup, $\gamma<_{e}\delta$ implies that $C_{\gamma}\subsetneq
C_{\delta}\subsetneq G_{2}$, and
$\bigcup_{\gamma\in\mathcal{B}}C_{\gamma}=G_{2}$. ∎

## References

  * [1] Steven Boyer, Cameron McA. Gordon, and Liam Watson.  On L-spaces and left-orderable fundamental groups.  Mathematische Annalen, 356(4):1213--1245, 2013.
  * [2] Steven Boyer, Dale Rolfsen, and Bert Wiest.  Orderable 3-manifold groups.  Annales de l’Institut Fourier, 55(1):243--288, 2005.
  * [3] Adam Clay, Colin Desmarais, and Patrick Naylor.  Testing bi-orderability of knot groups.  Canadian Mathematical Bulletin, 59(3):472–482, 2016.
  * [4] Adam Clay and Dale Rolfsen.  Ordered groups, eigenvalues, knots, surgery and L-spaces.  Mathematical Proceedings of the Cambridge Philosophical Society, 152(1):115–129, 2012.
  * [5] Adam Clay and Dale Rolfsen.  Ordered Groups and Topology.  American Mathematical Society, Providence, Rhode Island, 2016.
  * [6] Anthony E. Clement, Stephen Majewicz, and Marcos Zyman.  The Collection Process and Basic Commutators, pages 75--106.  Springer International Publishing, Cham, 2017.
  * [7] Paul Conrad.  Right-ordered groups.  Michigan Mathematical Journal, 6(3):267 -- 275, 1959.
  * [8] Benson Farb and Dan Margalit.  A Primer on Mapping Class Groups (PMS-49).  Princeton University Press, 2012.
  * [9] P. Hall.  A contribution to the theory of groups of prime-power order.  Proceedings of the London Mathematical Society, s2-36(1):29--95, 1934.
  * [10] O. Hölder.  Die Axiome der Quantität und die Lehre vom Mass.  Teubner, 1901.
  * [11] Jonathan Johnson.  Residual torsion-free nilpotence, bi-orderability, and two-bridge links.  Canadian Journal of Mathematics, page 1–64, 2023.
  * [12] Jonathan Johnson.  Residual torsion-free nilpotence, biorderability and pretzel knots.  Algebr. Geom. Topol., 23(4):1787--1830, 2023.
  * [13] Eiko Kin and Dale Rolfsen.  Braids, orderings, and minimal volume cusped hyperbolic 3-manifolds.  Groups Geom. Dyn., 12(3):961--1004, 2018.
  * [14] F. W. Levi.  Ordered groups.  Proceedings of the Indian Academy of Sciences - Section A, 16(4):256, Oct 1942.
  * [15] Peter A. Linnell, Akbar H. Rhemtulla, and Dale P.O. Rolfsen.  Invariant group orderings and Galois conjugates.  Journal of Algebra, 319(12):4891 -- 4898, 2008.
  * [16] Wilhelm Magnus.  Beziehungen zwischen gruppen und idealen in einem speziellen ring.  Mathematische Annalen, 111(1):259--280, 1935.
  * [17] Bernard Perron and Dale Rolfsen.  On orderability of fibred knot groups.  Mathematical Proceedings of the Cambridge Philosophical Society, 135(1):147–153, 2003.
  * [18] Takafumi Yamada.  A family of bi-orderable non-fibered 2-bridge knot groups.  Journal of Knot Theory and Its Ramifications, 26(04):1750011, 2017\.


# Black Hole Solutions as Topological Thermodynamic Defects

Shao-Wen Wei1,2 111E-mail<EMAIL_ADDRESS>Yu-Xiao Liu1,2 222E-mail:
<EMAIL_ADDRESS>Robert B. Mann3 333E-mail<EMAIL_ADDRESS>1Lanzhou
Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu
Province, School of Physical Science and Technology, Lanzhou University,
Lanzhou 730000, People’s Republic of China,
2Institute of Theoretical Physics $\&$ Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, People’s Republic of China,
3 Department of Physics & Astronomy, University of Waterloo, Waterloo, Ont.
Canada N2L 3G1

###### Abstract

In this Letter, employing the generalized off-shell free energy, we treat
black hole solutions as defects in the thermodynamic parameter space. The
results show that the positive and negative winding numbers corresponding to
the defects indicate the local thermodynamical stable and unstable black hole
solutions, respectively. The topological number defined as the sum of the
winding numbers for all the black hole branches at an arbitrary given
temperature is found to be a universal number independent of the black hole
parameters. Moreover, this topological number only depends on the
thermodynamic asymptotic behaviors of the black hole temperature at small and
large black hole limits. Different black hole systems are characterized by
three classes via this topological number. This number could help us in better
understanding the black hole thermodynamics, and further shed new light on the
fundamental nature of quantum gravity.

###### pacs:

04.70.Dy, 04.70.Bw, 05.70.Ce

Introduction— Black holes predicted by general relativity play a central role
in modern physics. Observations of binary black hole mergers by the LIGO/Virgo
Collaboration Abbott  and reconstruction of event-horizon-scale images of M87*
by the Event Horizon Telescope (EHT) Akiyama1  have opened new windows to
study the strong gravitational nature of black holes. Alternatively, seeking
underlying characteristic properties of black holes is also quite valuable,
since they can yield features that can be tested by further astronomical
observations.

One such approach is topology: by ignoring detailed structure, generic
properties of a system can be discerned. This approach has been extensively
employed to study many physical phenomena such as magnetic monopoles, the
quantum Hall effect, and superconductivity. A key concept is that of defects,
which are generally thought to be related to zero points of a field at
$\vec{x}=\vec{z}$,

$\displaystyle\phi(\vec{x})|_{\vec{x}=\vec{z}}=0,$ (1)

in a space, and which can reveal certain properties of field configurations.
In different dimensions, a defect can be a point charge, a string, or even a
domain wall. The most simple topological quantity associated with the zero
point of a field is its winding number. When equipped with it, we can
determine the nature of a system possessing defects.

Of particular recent interest has been a special vector constructed in the
coordinate space of a black hole spacetime Cunhaa ; Cunhab  by making use of
null geodesics. The light ring of a black hole is located exactly at the zero
point of this vector field. After calculating the winding number corresponding
to this zero point, a four-dimensional stationary nonextremal spinning black
hole in asymptotically flat spacetime with a topologically spherical Killing
horizon was shown to allow at least one standard light ring outside the
horizon for each sense of rotation Cunhab . This treatment was generalized to
other cases, where the critical points and timelike circular orbit were well
studied Wei2019 ; Guo ; Wei2112 ; Ahmed ; Yerra ; Bhamidipati ; Wei2022 .

In general relativity, a black hole solution satisfies the Einstein field
equations, which we reformulate as

$\displaystyle\mathcal{E}_{\mu\nu}\equiv G_{\mu\nu}-\frac{8\pi
G}{c^{4}}T_{\mu\nu}=0.$ (2)

We propose, analogous to Eq. (1), that a physical black hole solution is a
zero point of the tensor field $\mathcal{E}_{\mu\nu}$, at which all its
components vanish. Although the Einstein field equations also admit other
defect-like solutions, such as cosmic strings and branes, we here only focus
on black holes. As a result, we can endow a black hole solution with a
topological charge as well. Adopting this topological argument, we can then
study the local and global particular properties of a black hole. Different
black hole solutions are also expected to be divided into different classes.

Realization in thermodynamics— As a starting point to realize the idea that a
black hole solution possesses a topological charge, we begin with black hole
thermodynamics.

In 1977, Gibbons and Hawking Gibbons  proposed that the partition function of
a canonical ensemble for black holes can be evaluated with its Euclidean
action in the form of the gravitational path integral. In “zero-loop”
approximation, it reads

$\displaystyle\mathcal{Z}=e^{-\beta F}=\int
D[g]e^{-\frac{\mathcal{I}}{\hbar}}\sim e^{-\frac{\mathcal{I}}{\hbar}},$ (3)

where $F$ and $\mathcal{I}$ is the free energy and Euclidean action of the
black hole. The period $\beta$ of the Euclidean time is the inverse of the
black hole temperature. Following this approach, it was found that negative
heat capacity and imaginary energy fluctuations were produced. Soon afterward,
these shortcomings were solved by York York  by imagining that the black hole
is placed inside a cavity. Fixing the temperature $T$ of the cavity surface,
the results show that the heavy-mass black hole branch with mass
$M>\sqrt{3}/8\pi T$ is thermodynamically stable, and thus the corresponding
partition function is well defined.

In order to describe a black hole with an arbitrary mass, York defined a
generalized free energy following the standard definition of the free energy
with the mass and temperature being two independent variables York . This
naturally extends one more dimension for the thermodynamic parameter space of
the black hole. Furthermore, the generalized free energy reduces to the
standard one when the relation between the black hole mass and temperature is
satisfied Whiting .

Inspired by this, we would like to introduce the generalized free energy

$\displaystyle\mathcal{F}=E-\frac{S}{\tau},$ (4)

for a black hole system with energy $E$ and entropy $S$. The parameter $\tau$
is an extra variable having the dimension of time, and can be thought as the
inverse temperature of the cavity enclosing the black hole. Here we let the
time parameter $\tau$ vary freely instead of the mass York . In general this
generalized free energy is off-shell except at

$\displaystyle\tau=T^{-1},$ (5)

at which the black hole solution satisfies the Einstein field equations (2).
Recently, a similar generalized free energy (4) has been used to study the
dynamic evolution of black hole phase transitions Li ; Wang ; weiMann ; Mann .

We now describe the explicit construction for the vector $\phi$ in (1) via a
thermodynamic approach. We define

$\displaystyle\phi=\left(\frac{\partial\mathcal{F}}{\partial
r_{\text{h}}},-\cot\Theta\csc\Theta\right)$ (6)

where, inspired by the axis limit Cunhab , we introduce a parameter
$0\leq\Theta\leq\pi$ for convenience. At $\Theta=0$, $\pi$, the component
$\phi^{\Theta}$ diverges and thus the direction of the vector is outward.

More importantly, the zero points of $\phi$ correspond to the conditions
$\Theta=\pi/2$ and $\tau=T^{-1}$. This confirms that the black hole solution
exactly meets the zero point of the vector $\phi$. Therefore, from the
viewpoint of topology, we can endow each black hole solution with a
topological charge by using $\phi$.

Following Duan’s $\phi$-mapping topological current theory Duana ; Duanb , we
can introduce the topological current as

$\displaystyle
j^{\mu}=\frac{1}{2\pi}\epsilon^{\mu\nu\rho}\epsilon_{ab}\partial_{\nu}n^{a}\partial_{\rho}n^{b},\quad\mu,\nu,\rho=0,1,2,$
(7)

where $\partial_{\nu}=\frac{\partial}{\partial x^{\nu}}$ and
$x^{\nu}=(\tau,~{}r_{\text{h}},~{}\Theta)$. The unit vector is defined as
$n^{a}=\frac{\phi^{a}}{||\phi||}$ ($a=1,2$) with
$\phi^{1}=\phi^{r_{\text{h}}}$ and $\phi^{2}=\phi^{\Theta}$. Here the
parameter $\tau$ can be considered as a time parameter of the topological
defect. Moreover, it is easy to check that this topological current is
conserved, i.e., $\partial_{\mu}j^{\mu}=0$. By making use of the Jacobi tensor
$\epsilon^{ab}J^{\mu}\left(\frac{\phi}{x}\right)=\epsilon^{\mu\nu\rho}\partial_{\nu}\phi^{a}\partial_{\rho}\phi^{b}$
and the two-dimensional Laplacian Green function
$\Delta_{\phi^{a}}\ln||\phi||=2\pi\delta^{2}(\phi)$, the topological current
can be reexpressed as Wei2019

$j^{\mu}=\delta^{2}(\phi)J^{\mu}\left(\frac{\phi}{x}\right).$ (8)

Obviously, $j^{\mu}$ is nonzero only at $\phi^{a}(x^{i})=0$, and we denote its
$i$-th solution as $\vec{x}=\vec{z}_{i}$. The density of the topological
current is then Schouton

$j^{0}=\sum_{i=1}^{N}\beta_{i}\eta_{i}\delta^{2}(\vec{x}-\vec{z}_{i}).$ (9)

The positive Hopf index $\beta_{i}$ counts the number of the loops that
$\phi^{a}$ makes in the vector $\phi$ space when $x^{\mu}$ goes around the
zero point $z_{i}$, and the Brouwer degree
$\eta_{i}=\text{sign}(J^{0}({\phi}/{x})_{z_{i}})=\pm 1$. Given a parameter
region $\Sigma$, the corresponding topological number can be obtained

$\displaystyle
W=\int_{\Sigma}j^{0}d^{2}x=\sum_{i=1}^{N}\beta_{i}\eta_{i}=\sum_{i=1}^{N}w_{i},$
(10)

where $w_{i}$ is the winding number for the $i$-th zero point of $\phi$
contained in $\Sigma$. If two given loops $\partial\Sigma$ and
$\partial\Sigma^{\prime}$ enclose the same zero point of $\phi$, they possess
the same winding number. Alternatively, if there is no zero point in the
enclosed region, we will have $W=0$. If $\Sigma$ is the neighbourhood of a
zero point of $\phi$, it will yield local topological properties, whereas if
$\Sigma$ is the entire parameter space, the global topological $W$ number will
be revealed.

In the above approach, the isolated zero points require the Jacobian
$J_{0}(\phi/x)\neq 0$. If this condition is violated, the defect bifurcates Fu
.

Local and global topological properties—Following Eq. (3), one can obtain the
free energy by calculating the first-order Euclidean Einstein action

$\displaystyle\mathcal{I}=-\frac{1}{16\pi}\int\sqrt{g}Rd^{4}x+\frac{1}{8\pi}\oint\sqrt{\gamma}Kd^{3}x-\mathcal{I}_{subtract}$
(11)

where $K$ is the trace of the extrinsic curvature, $\gamma_{ij}$ is the
induced metric on the boundary, and $\mathcal{I}_{subtract}$ is the
subtraction term.

For the Schwarzschild black hole, one easily has
$E=\partial_{\beta}\mathcal{I}=M$ and
$S=(\beta\partial_{\beta}\mathcal{I}-\mathcal{I})=4\pi M^{2}$ Gibbons .
Accordingly, the black hole thermodynamic relations can be deduced
straightforwardly. For the Schwarzschild black hole, we obtain the generalized
free energy $\mathcal{F}=\frac{r_{\text{h}}}{2}-\frac{\pi
r_{\text{h}}^{2}}{\tau}$, where $r_{\text{h}}=2M$. Then the components of the
constructed vector $\phi$ can be calculated

$\displaystyle\phi^{r_{\text{h}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{2\pi r_{\text{h}}}{\tau},$ (12)
$\displaystyle\phi^{\Theta}$ $\displaystyle=$
$\displaystyle-\cot\Theta\csc\Theta.$ (13)

We show the unit vector field $n$ on a portion of the $\Theta$-$r_{\text{h}}$
plane in Fig. 1 for the Schwarzschild black hole with $\tau=4\pi r_{0}$ with
$r_{0}$ an arbitrary length scale set by the size of a cavity surrounding the
black hole.

From the figure, the zero point is located at $r_{\text{h}}/r_{0}=1$ and
$\Theta=\pi/2$. Since the winding number $w$ is independent of these loops
enclosing the zero point, we can calculate it for an arbitrary loop; for
example see $C_{1}$ given in Fig. 1. Performing the calculation, we obtain the
winding number $w=-1$. If we choose an alternate orientation convention by
adding a minus sign in $\phi^{r_{\text{h}}}$ in (6), we obtain $w=1$ instead.
However the winding numbers of other types of black holes will likewise be
changed.

Figure 1: The red arrows represent the unit vector field $n$ on a portion of
the $r_{\text{h}}$-$\Theta$ plane. The zero points (ZPs) marked with black
dots are at ($r_{\text{h}}$, $\Theta$)=(1, $\frac{\pi}{2}$), (1.46,
$\frac{\pi}{2}$), and (2.15, $\frac{\pi}{2}$), for ZP1, ZP2, and ZP3,
respectively. The blue contours $C_{i}$ are closed loops enclosing the zero
points. (a) The unit vector field for the Schwarzschild black hole with
$\tau/r_{0}=4\pi$. (b) The unit vector field for the RN black hole with
$\tau/r_{0}=34.48$ and $Q/r_{0}=1$.

Considering the gravitational action together with the electromagnetic field,
it is easy to find $E=\frac{r_{\text{h}}^{2}+Q^{2}}{2r_{\text{h}}}$ and $S=\pi
r_{\text{h}}^{2}$ for the Reissner-Nordström (RN) black hole. Thus the
generalized free energy is

$\displaystyle\mathcal{F}=\frac{r_{\text{h}}^{2}+Q^{2}}{2r_{\text{h}}}-\frac{\pi
r_{\text{h}}^{2}}{\tau}.$ (14)

Here $\tau$ can be thought as the inverse temperature of the cavity. Thus
$r_{\text{h}}$ is independent of $\tau$, and can vary freely, ignoring the
lower bound. Employing it, we plot the unit vector field $n$ in Fig. 1 for
arbitrarily chosen typical values $\tau/r_{0}=34.48$ and $Q/r_{0}=1$.

We find two zero points, ZP2 and ZP3, at $r_{\text{h}}/r_{0}=1.46$ and $2.15$
respectively. A detailed calculation shows that their respective winding
numbers are $w=1$ and $-1$. We also obtain the heat capacities
$C_{Q}/r_{0}^{2}=18.01$ and $-64.81$ at constant charge $Q/r_{0}=1$ for the
respective positive and negative zero points. Noting that each zero point of
the unit vector has a winding number $1$ or $-1$, from this we conjecture that
winding number is related to local thermodynamic stability, with
positive/negative values corresponding to stable/unstable black hole
solutions.

Turning to global properties of the topology, if we choose the region $\Sigma$
as the whole parameter space or the loop $\partial\Sigma$ as the boundary of
the parameter space, i.e., $(0<\Theta<\pi)\cup(0<r_{\text{h}}<\infty)$, we can
obtain the topological number $W$ for a black hole solution. This global
property can be used to classify different black hole solutions. Since we here
take the axis limit, where the direction of the vector $\phi$ is up at
$\Theta=\pi$ and down at $\Theta=0$, the value of the topological number $W$
actually depends on the direction of the vector at $r_{\text{h}}=0$ and
$\infty$. This suggests that black hole solutions sharing the same behavior at
$r_{\text{h}}=0$ and $\infty$ possess the same topological number, and thus
are topologically equivalent.

By solving the equation $\phi^{r_{\text{h}}}=0$, we can obtain a curve in the
$r_{\text{h}}-\tau$ plane. The results are

$\displaystyle\tau=4\pi r_{\text{h}}$ (15)

and

$\displaystyle\tau=\frac{4\pi r_{\text{h}}^{3}}{r_{\text{h}}^{2}-Q^{2}}$ (16)

for the Schwarzschild and RN black holes, respectively. To show zero points of
the component $\phi^{r_{\text{h}}}$, we plot these two curves in Fig. 2. For
large $\tau$ (e.g. $\tau=\tau_{2}$) there are respectively one and two
intersection points for the Schwarzschild and RN black holes. The intersection
points exactly satisfy the condition (5), and therefore denote the on-shell
black hole solutions with temperature $T=\tau^{-1}$. In contrast to the
Schwarzschild black hole, for $\tau<\tau_{c}$, the two intersection points for
the RN black hole coincide and then disappear. Based on the local property of
a zero point, we have the topological number $W=-1$ for the Schwarzschild
black hole, while $W=1-1=0$ for the RN black hole with the charge $Q/r_{0}=1$.
In particular, at the point $\tau_{c}=6\sqrt{3}\pi Q$, it is easy to get
$\frac{d^{2}\tau}{dr_{\text{h}}^{2}}=6\sqrt{3}\pi/Q>0$ for the RN black hole.
This suggests that $\tau_{c}$ is a generation point, which can also be
observed in Fig. 2.

To investigate whether or not $W$ depends on the charge, we examine the
behavior of the curve $\tau(r_{\text{h}})$ in Eq. (16) at the limit
$r_{\text{h}}\rightarrow r_{\text{E}}=Q$ (corresponding to extremal RN black
hole with smallest horizon) and $\infty$. Obviously, for a nonvanishing charge
$Q$, the vanishing/diverging behaviour of $\tau$ as $r_{\text{h}}\rightarrow
r_{\text{E}}/\infty$ does not change, and so $W$ remains the same for
different values of $Q$. For the Schwarzschild black hole with $Q=0$, the
behaviour of $\tau(r_{\text{h}})$ for small $r_{\text{h}}$ differs from the
charged case, and so the topological numbers $W$ for the two black hole
solutions are different.

Figure 2: Zero points of the vector $\phi$ shown in the $r_{\text{h}}$-$\tau$
plane. The blue dashed and red solid lines are for the Schwarzschild black
hole (Sch BH) and RN black hole (BH) with $Q/r_{0}=1$. The black dot with
$\tau_{c}=6\sqrt{3}\pi Q$ denotes the generation point for the RN black hole.
At $\tau=\tau_{1}$, there is one Schwarzschild black hole, and at
$\tau=\tau_{2}$, there are one Schwarzschild black hole and two RN black
holes. Figure 3: Zero points of $\phi^{r_{\text{h}}}$ shown in the
$r_{\text{h}}$-$\tau$ plane for the RN-AdS black hole with $Pr_{0}^{2}=0.0022$
and $Q/r_{0}=1$. The black solid, blue dashed, and red solid lines are for the
small black hole (SBH), intermediate black hole (IBH), and large black hole
(LBH), respectively. Black and blue dots are the annihilation and generation
points. Different color regions have different numbers of the black hole
branches. However their $W$ number is constant and equals 1.

Another interesting black hole solution is the charged Reissner-Nordström
anti-de Sitter (RN-AdS) black hole, which exhibits a small-large black hole
phase transition Kubiznak . In the extended phase space, the cosmological
constant is treated as the pressure $P$ of the system Kastor . The free energy
is

$\displaystyle\mathcal{F}=\frac{8\pi
Pr_{\text{h}}^{4}+3r_{\text{h}}^{2}+3Q^{2}}{6r_{\text{h}}}-\frac{\pi
r_{\text{h}}^{2}}{\tau}.$ (17)

Taking the pressure $Pr_{0}^{2}=0.0022$ to be smaller than its critical value,
we exhibit zero points of $\phi^{r_{\text{h}}}$ in the $r_{\text{h}}-\tau$
plane in Fig. 3 with $Q/r_{0}=1$. Quite different from the Schwarzschild and
RN black holes shown in Fig. 2, we observe that there are three black hole
branches for $\tau_{a}<\tau<\tau_{b}$, one small black hole branch for
$\tau<\tau_{a}$ and one large black hole branch for $\tau>\tau_{b}$. Computing
the winding number for these black hole branches, we find that the small and
large black hole branches have $w=1$, while the intermediate black hole branch
has $w=-1$. Therefore, at this pressure the RN-AdS black hole always has
$W=1-1+1=1$, which is independent of $\tau$. Since the pressure $P$ is
positive for the RN-AdS black hole, it does not affect the asymptotic behavior
of $\tau$ at small and large $r_{\text{h}}$. As a result, the topological
number $W$ is always unity for the RN-AdS black hole. Furthermore, our result
that thermodynamically stable small and large black holes have $w=1$ and
unstable intermediate black holes have $w=-1$ supports our conjecture that
positive/negative winding number is indicative of thermodynamic
stablilty/instability.

Note that for these values of $Pr_{0}^{2}$ and $Q/r_{0}$, one generation point
and one annihilation point can be found at $\tau/r_{0}=\tau_{a}/r_{0}=25.84$
and $\tau/r_{0}=\tau_{b}/r_{0}=27.62$, respectively. If the pressure is larger
than its critical value, $r_{\text{h}}/r_{0}$ will be a monotonically
increasing function of $\tau/r_{0}$, and thus no bifurcation phenomenon will
be observed. However the topological number $W$ still stays the same.

| Sch BH | RN BH | RN-AdS BH
---|---|---|---
$W$ | -1 | 0 | 1
generation point | 0 | 1 | 1 or 0
annihilation point | 0 | 0 | 1 or 0
Table 1: The topological number $W$, numbers of annihilation and generation
points for the Schwarzschild, RN, and RN-AdS black holes.

In the Supplemental Material we show that $d$-dimensional RN-AdS black holes
also have topological number $W=1$, consistent with the $d=4$ case (see
Supplemental Material).

Conclusions— Summarizing our results in Table 1, we have constructed a
universal topological number $W$ by treating the black hole solution as a
defect from the viewpoint of thermodynamics. Different black hole solutions
are characterized by different topological numbers and belong to different
topological classes.

Locally, each zero point of the vector field defined in (6) with the
generalized free energy exactly corresponds to one on-shell black hole
solution. Thus each black hole solution is endowed with one winding number.
Our study shows that the thermodynamic stability of the black hole is
indicated from the value of the winding number. Positive/negative winding
number corresponds to thermodynamically stable/unstable black hole solutions.
Of particular interest are the annihilation and generation points, which may
be quite important for the time evolution of a black hole placed in a cavity.

When considering the full parameter space, the topological number $W$ provides
us with a global topological property of a black hole solution. We have shown
that the Schwarzschild black hole, RN black hole, and RN-AdS black hole have
respectively $W=-1,~{}0$, and $1$, independent of the black hole parameters.
Since the topological number $W$ is the sum of the winding numbers for the
zero points of the vector, and it is only dependent on the behavior of the
curve $\tau(r_{\text{h}})$ at small and large $r_{\text{h}}$ limits, we
conjectured that the topological number $W$ can take three values, $-1$, $0$,
and $1$. This suggests that other black hole solutions shall be divided into
three characteristic topological classes, a conjecture in need of further
confirmation.

We close by considering whether geometric modifications to black holes, such
as those induced by scalar hair, affect the topological number $W$. In
contrast to global charges (primary hair), black holes possessing new non-
trivial fields (secondary/pseudo hair) have long been of interest in string
theory Campbell ; Kaloper ; Kanti  and have attracted much recent attention in
both GR and modified gravity theories Herdeiro ; Herdeirorev ; Silva ; Radu ;
Cunha ; Collodel ; Zhang ; Konoplya . For the Einstein-Maxwell-scalar model
Radu ; Konoplya , we show in the Supplemental Material that the topological
number $W=0$ for these scalarized black holes, the same as that of the RN
black holes, and in support of the expectation that secondary (scalar) hair
does not change $W$ despite the fact that it modifies the free energy of a
black hole (see Supplemental Material). In more general string-theoretic
settings (for example axion fields with Lorentz Chern-Simons coupling to
gravity Campbell ) the generalized free energy will be modified. Investigation
of the topological charge for these more general black holes with hair is an
interesting project for further study.

In conclusion, our topological approach classifies each black hole solution
into certain classes sharing the similar thermodynamic properties. It provides
a considerable material for the topology of black hole thermodynamics. We
expect to uncover the deeper nature of other black hole solutions via the
topological approach.

_Acknowledgements_.—This work was supported by the National Natural Science
Foundation of China (Grants No. 12075103, No. 11875151, and No. 12047501), the
111 Project (Grant No. B20063), and the National Key Research and Development
Program of China (Grant No. 2020YFC2201503) and by the Natural Sciences and
Engineering Research Council of Canada.

Supplemental Material A: Higher dimensional black holes

The bulk action describing the $d$-dimensional charged Reissner-Nordström (RN)
AdS black hole solution reads

$\displaystyle\mathcal{I}=-\frac{1}{16}\int
d^{d}x\sqrt{-g}\left(R-F^{2}+\frac{(d-1)(d-2)}{l^{2}}\right).$ (18)

Solving the resultant Einstein-Maxwell equations we obtain the black hole
metric

$\displaystyle ds^{2}=-fdr^{2}+f^{-1}dr^{2}+r^{2}d\Omega^{2}_{d-2},$ (19)
$\displaystyle F=dA,\quad A=-\sqrt{\frac{d-2}{2(d-3)}}\frac{q}{r^{d-3}}dt$
(20)

where

$\displaystyle
f=1-\frac{m}{r^{d-3}}+\frac{q^{2}}{r^{2(d-3)}}+\frac{r^{2}}{l^{2}}.$ (21)

The Arnowitt-Deser-Misner mass $M$ and charge $Q$ of the black hole,
respectively, are

$\displaystyle M=\frac{d-2}{16\pi}\omega_{d-2}m,\quad
Q=\frac{\sqrt{2(d-2)(d-3)}}{8\pi}\omega_{d-2}q,$ (22)

with $\omega_{d-2}=2\pi^{(d-1)/2}/\Gamma((d-1)/2)$ being the volume of the
unit ($d-2$)-sphere. In the extended phase space, one has the thermodynamic
pressure $P=\frac{(d-1)(d-2)}{16\pi l^{2}}$. For fixed charge the total action
can be calculated by adding a boundary term Chamblin

$\displaystyle\mathcal{\tilde{I}}=\mathcal{I}-\frac{1}{4\pi}\int
d^{d-1}x\sqrt{h}F^{\mu\nu}n_{\mu}A_{\nu},$ (23)

where $n_{\mu}$ is a radial unit vector pointing outwards.

For the black hole solution (21), a computation of the action yields the free
energy $F=\mathcal{\tilde{I}}/\beta=M-TS$. Using the topological approach we
proposed, we obtain

$\displaystyle\tau=\frac{4\pi(d-2)\omega_{d-2}^{2}r_{\text{h}}^{2d+1}}{\omega_{d-2}^{2}r_{\text{h}}^{2d}\left((d-5)d+16\pi
Pr_{\text{h}}^{2}+6\right)-32\pi^{2}Q^{2}r_{\text{h}}^{6}}$ (24)

as the zero point of the vector field.

As a specific example, we take $d=6$, $Q/r_{0}^{3}=1$, and $Pr_{0}=0.1$, and
show the zero points of $\phi$ in the $r_{\text{h}}$-$\tau$ plane in Fig. 4.
Obviously, the behavior is similar to the four-dimensional black hole case
exhibited in Fig. 3 in the main text. The result indicates that the
topological number $W=1$ for the six-dimensional RN-AdS black hole. In order
to examine whether the topological number has any dimension dependence, we
respectively expand $\tau$ at small and large $r_{\text{h}}$, obtaining

$\displaystyle\tau\sim\infty\quad\text{for}\quad r_{\text{h}}\rightarrow
r_{\text{E}},$ (25) $\displaystyle\tau\sim\frac{(d-2)\pi}{4\pi
Pr_{\text{h}}}\quad\text{for}\quad r_{\text{h}}\rightarrow\infty,$ (26)

where $r_{\text{E}}$ denotes the horizon radius of the extremal black hole
with zero temperature and qualtitatively the same as Eqs. (15) and (16) in the
main text. Hence the topological number is the same as the $d=4$ RN-AdS black
hole, and the spacetime dimension $d$ thus has no effect on the topological
number. We conclude that all RN-AdS black holes have $W=1$, and belong to the
same topological class.

Furthermore, in any dimension small and large charged AdS black holes are
thermodynamical stable, whereas their intermediate counterparts are unstable.
In order to obtain $W=1$, the small and large black hole must have winding
number $+1$ and intermediate one has $-1$, which uniquely gives $W=1+1-1=1$ as
expected. A simple calculation will also show the same result. This further
indicates that thermodynamical stable or unstable black hole has positive or
negative winding number.

Figure 4: Zero points of $\phi^{r_{\text{h}}}$ shown in the
$r_{\text{h}}$-$\tau$ plane for the $d=6$-dimensional RN-AdS black hole with
$Pr_{0}^{2}=0.1$ and $Q/r_{0}^{3}=1$. The black solid, blue dashed, and red
solid lines are for the small, intermediate, and large black hole branches,
respectively. This pattern is quite similar to the $d=4$-dimensional case.

Supplemental Material B: Scalarized black holes

Here we investigate the effect of black hole scalar hair on the topological
number $W$.

To this end we consider the following Einstein-Maxwell-scalar model Radu ;
Konoplya

$\displaystyle S=\int
d^{4}\sqrt{-g}\left(R-2g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-f(\phi)F_{\mu\nu}F^{\mu\nu}\right),$
(27)

whose field equations yield different spherically symmetric scalarized black
holes for different scalar functions $f(\phi)$. For the ansatz

$\displaystyle
ds^{2}=-N(r)e^{-2\delta(r)}dt^{2}+\frac{dr^{2}}{N(r)}+r^{2}d\Omega^{2}_{2},$
(28)

we must have

$\displaystyle\lim_{r\rightarrow\infty}N(r)=1,\quad\lim_{r\rightarrow\infty}\delta(r)=0,$
(29)

assuming that the metic is asymptotically Minkowskian and that time is
measured by the coordinate $t$. We further assume that such scalarized black
holes are of the RN type, meaning that there exists a lower bound of the
horizon radius, at which the temperature of the black hole vanishes.

In order to evaluate the free energy of the black hole, we substitute the back
hole solution for the action (27), including both boundary and subtraction
terms. It is clear that the free energy is modified by the scalar, and so
scalarized black holes exhibit thermodynamic behaviour different from their RN
counterparts.

Despite this, we can provide some results about the topological number for
these scalarized black holes. The zero point of the vector constructed in our
topological approach can be written as

$\displaystyle\tau=\frac{4\pi
r_{\text{h}}e^{\delta(r_{\text{h}})}}{1-\frac{Q^{2}}{f(p_{0})r_{\text{h}}^{2}}},$
(30)

where $Q$ is the black hole charge and $p_{0}>0$ is the value of scalar field
at the horizon. The denominator of $\tau$ is nonnegative, and vanishes for the
extremal scalarized black hole with vanishing temperature. So similar to the
RN black hole, near the lower bound of the horizon radius, we have

$\displaystyle\tau\sim\infty\quad\text{for}\quad r_{\text{h}}\rightarrow
r_{\text{E}}.$ (31)

Making use of (29), one knows $e^{\delta(r_{\text{h}})}\rightarrow 1$ for
large $r_{\text{h}}$. Therefore, we have

$\displaystyle\tau\sim 4\pi r_{\text{h}}\quad\text{for}\quad
r_{\text{h}}\rightarrow\infty.$ (32)

This result is exactly consistent with Eqs. (15) and (16) in the main text for
the RN black hole. Hence the topological charge $W=0$ for this class of
scalarized black holes. Of notable interest is that the scalar function
$f(\phi)$ makes no contribution to the asymptotic behavior of $\tau$.

Summarizing, in this Einstein-Maxwell-scalar model, the scalar field does not
change the topological charge $W$. However, the existence of more than one
pair stable and unstable scalarized black hole branches remains a possibility,
and so the sum of the local winding numbers $w$ could nontrivially yield
$W=0$.

## References

  * (1) B. P. Abbott et al (Virgo, LIGO Scientific), Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016), [arXiv:1602.03837 [gr-qc]].
  * (2) K. Akiyama _et al_. (Event Horizon Telescope), First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole, Astrophys. J. 875, L1 (2019), [arXiv:1906.11238 [astro-ph.GA]].
  * (3) P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, Light ring stability in ultra-compact objects, Phys. Rev. Lett. 119, 251102 (2017), [arXiv:1708.04211 [gr-qc]].
  * (4) P. V. P. Cunha and C. A. R. Herdeiro, Stationary Black Holes and Light Rings, Phys. Rev. Lett. 124, 181101 (2020), [arXiv:2003.06445 [gr-qc]].
  * (5) S.-W. Wei, Topological charge and black hole photon spheres, Phys. Rev. D 102, 064039 (2020), [arXiv:2006.02112 [gr-qc]].
  * (6) M. Guo and S. Gao, Universal Properties of Light Rings for Stationary Axisymmetric Spacetimes, Phys. Rev. D 103, 104031 (2021), [arXiv:2011.02211 [gr-qc]].
  * (7) S.-W. Wei and Y.-X. Liu, Topology of black hole thermodynamics, Phys. Rev. D 105, 104003 (2022) [arXiv:2112.01706 [gr-qc]].
  * (8) M. B. Ahmed, D. Kubiznak, and R. B. Mann, Vortex/anti-vortex pair creation in black hole thermodynamics, [arXiv:2207.02147 [hep-th]].
  * (9) P. K. Yerra and C. Bhamidipati, Topology of black hole thermodynamics in Gauss-Bonnet gravity, Phys. Rev. D 105, 104053 (2022), [arXiv:2202.10288 [gr-qc]].
  * (10) P. K. Yerra and C. Bhamidipati, Topology of Born-Infeld AdS black holes in 4D novel Einstein-Gauss-Bonnet gravity, [arXiv:2207.10612 [gr-qc]].
  * (11) S.-W. Wei and Y.-X. Liu, Topology of equatorial timelike circular orbits around stationary black holes, [arXiv:2207.08397 [gr-qc]].
  * (12) G. W. Gibbons and S. W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15, 2752 (1977).
  * (13) J. W. York, Black-hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D 33, 2092 (1986).
  * (14) B. F. Whiting, Black holes and gravitational thermodynamics, Class. Quantum Grav. 7, 15 (1990).
  * (15) R. Li and J. Wang, Thermodynamics and kinetics of Hawking-Page phase transition, Phys. Rev. D 102, 024085 (2020).
  * (16) R. Li, K. Zhang, and J. Wang, Thermal Dynamic Phase Transition of Reissner-Nordstrom Anti-de Sitter Black Holes on Free Energy Landscape, J. High Energ. Phys. 2020, 090 (2020), [arXiv:2008.00495 [hep-th]].
  * (17) S.-W. Wei, Y.-Q. Wang, and Y.-X. Liu, Dynamic properties of thermodynamic phase transition for five-dimensional neutral Gauss-Bonnet AdS black hole on free energy landscape, Nucl. Phys. B 976, 115692 (2022), [arXiv:2009.05215 [gr-qc]].
  * (18) S.-W. Wei, Y.-Q. Wang, Y.-X. Liu, and R. B. Mann, Observing dynamic oscillatory behavior of triple points among black hole thermodynamic phase transitions, Sci. China Phys. Mech. Astron. 64, 270411 (2021), [arXiv:2102.00799 [gr-qc]].
  * (19) Y. S. Duan and M. L. Ge, $SU$ (2) gauge theory and electrodynamics of $N$ moving magnetic monopoles, Sci. Sin. 9, 1072 (1979).
  * (20) Y. S. Duan, The structure of the topological current, SLAC-PUB-3301, (1984).
  * (21) J. A. Schouton, Tensor analysis for Physicists (Claredon, Oxford, 1951).
  * (22) L.-B. Fu, Y.-S. Duan, and H. Zhang, Evolution of the Chern-Simons vortices, Phys. Rev. D 61, 045004 (2000), [arXiv:hep-th/0112033].
  * (23) D. Kubiznak and R. B. Mann, $P$-$V$ criticality of charged AdS black holes, J. High Energy Phys. 1207, 033 (2012), [arXiv:1205.0559[hep-th]].
  * (24) D. Kastor, S. Ray, and J. Traschen, Enthalpy and the mechanics of AdS black holes, Class. Quant. Grav. 26, 195011 (2009), [arXiv:0904.2765[hep-th]].
  * (25) A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myerso, Charged AdS Black Holes and Catastrophic Holography, Phys. Rev. D 60, 064018 (1999), [arXiv:hep-th/9902170].
  * (26) B. A. Campbell, M. J. Duncan, N. Kaloper, and K. A. Olive, Axion hair for Kerr black holes, Phys. Lett. B 251, 34 (1990).
  * (27) B. A. Campbell, N. Kaloper, and K. A. Olive, Classical hair for Kerr-Newman black holes in string gravity, Phys. Lett. B 285, 199 (1992).
  * (28) P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley, Dilatonic black holes in higher curvature string gravity, Phys. Rev. D 54, 5049 (1996), [arXiv:hep-th/9511071].
  * (29) C. A. R. Herdeiro and E. Radu, Kerr black holes with scalar hair, Phys. Rev. Lett. 112, 221101 (2014), [arXiv:1403.2757 [gr-qc]].
  * (30) C. A. R. Herdeiro and E. Radu, Asymptotically flat black holes with scalar hair: a review, Int. J. Mod. Phys. D 24, 1542014 (2015), [arXiv:1504.08209 [gr-qc]].
  * (31) H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou, and E. Berti, Spontaneous scalarization of black holes and compact stars from a Gauss-Bonnet coupling, Phys. Rev. Lett. 120, 131104 (2018), [arXiv:1711.02080 [gr-qc]].
  * (32) C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual, and J. A. Font, Spontaneous scalarisation of charged black holes, Phys. Rev. Lett. 121, 101102 (2018), [arXiv:1806.05190 [gr-qc]].
  * (33) P. V. P. Cunha, C. A. R. Herdeiro, and E. Radu, Spontaneously scalarised Kerr black holes, Phys. Rev. Lett. 123, 011101 (2019), [arXiv:1904.09997 [gr-qc]].
  * (34) E. Berti, L. G. Collodel, B. Kleihaus, and J. Kunz, Spin-induced black hole scalarization in Einstein-scalar-Gauss-Bonnet theory, Phys. Rev. Lett. 126, 011104 (2021), [arXiv:2009.03905 [gr-qc]].
  * (35) C.-Y. Zhang, Q. C., Y. Liu, W.-K. Luo, Y. Tian, and B. Wang, Critical phenomena in dynamical scalarization of charged black hole, Phys. Rev. Lett. 128, 161105 (2022) , [arXiv:2112.07455 [gr-qc]].
  * (36) R. A. Konoplya and A. Zhidenko, Analytical representation for metrics of scalarized Einstein-Maxwell black holes and their shadows, Phys. Rev. D 100, 044015 (2019), [arXiv:1907.05551 [gr-qc]].