Special classes of algebraic integers in low-dimensional topology (Algebraic number theory and related topics)

Special classes

of

algebraic

integers

in

low-dimensional

topology

Eriko

Hironaka

April 15,

2005

Abstract

This note describes someopen problems concerning distributions ofspecial classes ofreal

algebraic integers such as algebraicunits, andSalem,P-V and Perron numbers. These special

algebraic integers appear naturally as geometric invariants in $\mathrm{l}\mathrm{o}\mathrm{w}rightarrow \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}$topology, We

relate properties of Salem, P-V and Perron to minimization problems in various geometric settings.

1

Introduction

A complex number $\alpha\in \mathbb{C}$is an algebraic integer if it is

a

root of a monic integerpolynomial. Two algebraic integers a and $\beta$

are

algebraically conjugat\^e vxritterr $\alpha\sim\beta$, it a and

$\beta$ satisfy thesame

irreduciblemonicintegerpolynomial. An algebraic integer$\alpha$is an algebraic unit if

$\alpha\sim\alpha^{-1}$.

Let $\alpha$ beareal algebraic integer witha $>1$. Consider all

$\beta\sim$a such that $\beta\neq\alpha$:

(i) it $|\beta|<|\alpha|$,then

a

is

a

Perron number’

(ii) if $|\beta|<1$, then $\alpha$ is aP-$V$number, and

(iii) if $|\beta|\leq 1$ with atleast

one

$|\beta|=1$, then $\alpha$ is a Salem number.

In this short note, we review definitions and known results concerning distributions of P-V, Salem and Perronnumbers (Section 2), and relatethem to geometric invariantsin

low-dimensional

topology, including lengths of geodesies, growth rates of automatic groups, and homological and geometric dilatations of surface homeomorphisms (Section 3).

2

Distributions of algebraic

integers

and

Lehmer’s problem

Let $P$ bethe set ofmonicinteger polynomials. Given $f\in P$, let $s_{f}$ be theset ofcomplexroots of

$f$ countedwithmultiplicity, and let $S^{+}(f)\subset s_{f}$ be the subsetof points outside the unit circle $C$

.

For $f\in P$ define

$N(f)$ $=$ $|\mathrm{S}^{+}(f)|$;

$\lambda(f)$ $=$ $\max\{|\alpha| : \alpha\in S(f)\}$; and

$M(f)$ $=$ $\prod$ $|\alpha|$

.

$\alpha\in \mathrm{S}^{+}(J)$

Hereanempty product isdefined toequal 1. Thenumber $M(f)$, also

an

algebraicinteger, iscalled

the Mahler

measure

of$f$.

The minimal polynomial for aroot ofunityis called a cyclotomicpolynomial. Thefollowingare

(i) $f$is aproduct of cyclotomicpolynomials;

(ii) $N(f)=0$ ; (iii) A(f) $=1$; and (iv) $M(f)=1$

.

Thus, $N(f)\mathrm{X}(\mathrm{f})$ and $M(f)$ can be considered as measures ofhow far$f$ isfrom beinga product

of cyclotomic polynomials. Let$\mathcal{T}\subset P$ hethe subset of products of cyclotomic polynomials. While $N(f)$ takes discrete values and $\lambda(f)$

can

get arbitrarilyclose to one from above it is not knownwhetherthereis alower bound for Mahlermeasuresgreater thanone. In 1933Lehmer [Leh]

posedthefollowing problem.

Question 1 (Lehmer’s problem) Given$\delta>0$, does there exist a

f

$\in P$ such that $1<M(f)<$

$1+\delta^{\varphi}$

It is not hard to see that for $f\in P$$\backslash \mathcal{T}$, ifwe fixthe degree $d$ of$f$

} then $\lambda(f)$ and $M(f)$ are

bounded from below by anumber greater than onedepending on $d$

.

Up to degree 40 there is no non-cyclotomic polynomial with Mahler

measure

less than that of

Lehmer’s candidate polynomial

$f_{L}(x)=x^{10}+x^{9}$–$x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1$

(see, for example, [Boydl] [Mos]), TheMahler

measure

$M(f_{L})$ is approximately 1.7628.

By aresult of Smyth in 1970, Lehmer’s problem reduces to the caseofreciprocal polynomials,

whichwe describe inSection 2.1. Section 2.2 givessome known results concerning distributions of Perron, Salem andP-Vnumbers.

2.1

Reciprocal

polynomials

Given $f\in P$of degree $d$, the reciprocal_{$f_{*}(x)$} of_$f(x)$ is defined tobe

$f_{*}(x)=x^{d}f(1/x)$.

A polynomial is reciprocal if$f$ .

$=f_{*}$. Visually,

a

reciprocal polynom ial is

one

for which the

coeffi-cients are palindromic, thatis, theyarethesamewritten from rightto left orlefttoright. Lehmer’s polynomial $f_{L}$ is areciprocal polynomial.

It$f$. satisfies $f=-f_{*}$, it is called anti-reciprocal A polynomial $f$ is anti-reciprocal if andonly

if $f(x)=(x-1)g(x)$ where $g(x)$ is reciprocal. All cyclotomic polynomials are reciprocal except

(x–1). A polynom ial is reciprocal or anti-reciprocal ifand only if it is a product of irreducible reciprocal polynomials and (x–1). A separable polynomial is reciprocal

or

anti-reciprocalif and

onlyif $S(f)$ _isclosed under inverses. Thus, an algebraic integer $\alpha$ is an algebraic unit if and only if its minimal polynomiai is reciprocal. An irreducible polynomial with aroot onthe unit circleis

automatically reciprocal. Thus, minimal polynomials ofSalem numbers are always reciprocal, and

theminimal polynomial ofa P-Vnumber is reciprocal only ifit is quadratic.

Smythshowed [Smy] that if$f\neq\pm f_{*}$, then the smallest Mahlermeasure is realized by

$f_{S}(x)=x^{3}-x-1$,

which hasMahler

measure

$M(fs)\approx$

1.32472.

SinceLehmer’s polynomial $f_{L}$ satisfies

$M(f_{L})<M(fs)$,

2.2

P-V

and Salem

polynomials

An interestingspecial caseofLehmer’s problemis when $N(f)$ $=1$. Thefollowingare equivalent:

(i) $N(f)=1$;

(ii) $f$ has asingleroot outside $C$, whichis (up to sign) a Salem number or a P-Vnumber; and

(iii) $f=gh$where$g\in \mathcal{T}$and $h$ istheminimal polynomial ofaSalem number or a P-V number. For quadratic polynomials $N(f)=1$ implies that both roots are real. The reciprocalcase is

discussedinSection 3.1. Forirreduciblepolynomialsof degree$>2$,$N(f)=1$ im lies $f$hasaSalem

root (up tosign) if and onlyif $f$is reciprocal oranti-reciprocal.

Theset ofP-Vnumbers is closed [Sal], and thesmallestaccumulation pointis the goldenmean

$\alpha_{G}$ (cf. Section 3.1). A complete set ofP-V numbersless than 1.6 was catalogued by Dufresnoy

andPisot [DP].

The polynomial $fs(z)=x^{3}-$$x$–1 isthe minimalpolynomialfor the smallest P-V number $\theta 0$

[Sie], and $f_{L}$ is the minimalpolynomial for the smallest known Salem number $\alpha_{L}$

.

It is an open

problem whether thereis alower bound larger thanone for the set of Salemnumbers, orwhether

there is aSalem number less than $f_{L}$.

In their study of distributions of Salem numbers, Salem [Sal] and Boyd [Boyd2] investigated sequences polynomials of the form

$Q_{n}(t)=t^{n}P(t)\pm P_{*}(t)$, (1)

for $P\in$ V. The sequence ofpolynom ials ofthe form givenin (1) is called a Salem-Boyd sequence

for $P$. Salem [Sal] proved that the set ofP-V numbers liesin the set of upper and lower limits of

Salem numbers byprovingthe following result.

Theorem 1 Given any P-$V$polynomial$\mathrm{P}$, let$Q_{n}$ beaSalem-Boydsequence

for

P. Then

for

some

$N>0$, $N\{Qn$) $=0$jar $n<N_{\}}$ and$N\{Qn$) $=1$

for

$n\geq N$. Fur thermore,

for

$n>N$, the Salem

numbers$M(Q_{n})=\lambda(Q_{n})$ converge monotonicallyto $M(P)=\lambda(P)$

from

above or below depending

on the sign.

Inthe more general setting where P $\in P$ is anyelement, Boyd showed the following [Boyd2],

Theorem 2 Let$Q_{n}$ be a Salem-Boyd sequence

for

a

monicintegerpolynomial$P(t)$

.

Then$\prime n$)$e$ have the following.

(1) $N(Q_{n})\leq N(P)$

for

all $n$ $\geq 1$;

(2) $\lim_{narrow\infty}\mathrm{N}\{\mathrm{Q}\mathrm{n}$) $=\lambda(P1j$ and

(3) $\lim_{narrow\infty}M(Q_{n})=M(P)$.

Anyreciprocal polynomial

can

be written inthe form of$Q_{n}$ for

some

$P$ and $n$. Thus, although

Theorem 2doesn’t givea lower boundon$M(Qn)$or$\mathrm{X}(\mathrm{Q}\mathrm{n})$ in terms of$M(P)$ and $\lambda(P)$,itdoes

par-titionthe setof Mahler measures and radiiofreciprocalpolynomials into (non-disjoint) convergent

families.

A polynomial $f$ is a Perron polynomial if there is asimple realroot $\alpha>1$, such that for any

otherroot $\beta$of $f$, $|\beta|<\alpha$. Lehmer’s problemis unsolved for thisspecial subclass of monic integer

polynomials. By definition, the characteristic polynomial of a $\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{n}rightarrow \mathrm{F}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{u}\mathrm{s}$

matrix is Perron

[Gan].

Theorem 3

_If

P is a Perron polynomial and$Q_{n}$ is a Salem-Boyd sequence

for

P. Then$\lambda(Q_{n})$ is

an eventuallymonotone sequence convergeng to $\lambda(P)$,

In general, $M(Q_{n})$ is not monotone, eventually monotone, or monotone for an arithmetic

sub-sequence

3

Examples

from low-dimensional topology and geometry

In this section, wclist some occurrencesofreciprocal, Salem, P-V and Perron polynomials in low

dimensional topology. These exam ples indicate a common underlying structure behind many of

the invariants of low dimensional topology, whichis yet to be fully explored.

3.1

Quadratic

polynomials

There is a bijective correspondence between $\Gamma=\mathrm{S}\mathrm{L}(2, \mathbb{Z})$ and reciprocal quadratic polynomials. This is defined by

$A\mapsto f_{A}$,

where $f_{A}$ is the characteristic polynomial for $A\in\Gamma$

.

The characteristic polynomial $f_{A}$ of any element $A\in\Gamma$ isreciprocal, since the twoeigenvaluesof $A$must multiply to 1.

The inverse map is defined as follows. Let A be a quadratic such that $a=\lambda+1/\lambda\in \mathbb{Z}$, and

define

A $=($ $-1a$ $01$

).

ThenA and$\lambda^{-1}$

are

the roots of_{$f_{A}$}

.

Thecorrespondence

$\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)\mapsto$A$(fA)$ (2)

is orderpreserving, and $\lambda(f_{A})=1$ ifand only if $|’\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{e}(A)|\leq 2$. Thus, non-cyclotomic reciprocal

quadraticscorrespond to hyperbolic elements of$\Gamma$

.

Consider the action of$\Gamma$ as isometries on the hyperbolic disk $\mathbb{H}^{2}$. Then hyperbolic elements

$A\in\Gamma$ correspondto closed geodesies $\gamma_{A}$ on the quotient space

$\Gamma_{\backslash }^{\backslash }\mathbb{H}^{2}$, whoselength $\ell(\gamma A)$ is given

by

$\ell(\gamma_{A})=\log(’\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{e}(A))$.

The correspondence givenin (2) also gives rise to anordering preserving correspondence between $\lambda(f_{A})$ andlengths of closedgeodesies

on

$\Gamma\backslash \mathbb{H}^{2}$

.

Iffollows that for quadratic reciprocal polynomials $f$, the smallest Mahler

measure

greater than

one

that

occurs

isrealized by

$f_{0}(x)=x^{2}-3x+1$

and$M(fo)$ $=\lambda(J_{0}^{\cdot})=(3 +\sqrt{5})/2$.

Remark4 There $\mathrm{o}s$asimilar correspondence betweenSalem numbers andlengths

of

closedgeodesies

on

more

general arithmetic quotients

_of

$\mathbb{H}^{2}$ (see,

for

example, [G-H]). Thus, the minimization

problem

_for

Salem numbers is related to the problem

_of

finding a minimum length geodesic on an

The smallest Mahler measure greater than one among all quadratics is realized by the non-reciprocal polynom ial

$f_{g}(x)=x^{2}-$_$x-1$

.

Here, $M\langle f_{\mathit{9}}$) $=(1+\sqrt{5})/2$ is the golden mean. The smallest Mahler measures greater than one

for reciprocal polynomials ofdegrees 4and 6

are

approximately 1.72208 and 1.40127 respectively,

and hence are also larger than the smallest non-reciprocal Mahlermeasure $M(fs)$. For degrees 8

and higher there always exists a reciprocal polynomial (not necessarily irreducible) with Mahler

measure

greater than one andlessthan$M(fs)$.

3.2

Transformations preserving lattices

Let $B$ be anon-degenerate symmetric bilinear form

on

$\mathbb{R}^{n}$, and suppose $M\in \mathrm{S}\mathrm{L}(n, \mathbb{Z})$ preserves

$B$

.

Equivalently, $M$preserves thelatticein$\mathbb{R}^{n}$definedby the innerproductassociatedto $B$

.

Then

the set of eigenvalues of $M$ is closed under inverses. Thus, iffor

exam

plc $M$ has no repeated

eigenvalues, orequivalently the characteristic polynomial$f_{M}$ is separable,then$f_{M}$ is reciprocal

or

anti-reciprocal (see,for example, [G-Mc], Theorem 2.1).

Consider the Coxeter element of

a

Coxeter system (see [Hum] for definitions). The Coxeter

element preserves

an

associatedsymmetric bilinear form definedby the Coxeter system. If $(W, S)$

isanirreducibleCoxeter system, and$\mathrm{f}(\mathrm{w},\mathrm{s})$isthe characteristic polyonomial of the Coxeterelement

of $(W, S)$, then$\lambda(f_{(W,S)})=1$ifandonly if$(W, S)$issphericaloraffine $[\mathrm{A}’ \mathrm{C}]$

.

If$(W, S)$ isirreducible

andnot spherical or affine, then $\lambda(f_{(W,S)})$ is minimized by the $B_{10}$ Coxeter system $(W0, S_{\mathit{0}})$, and

$f_{(W_{0},S_{0})}=f_{L}$ is Lehm cr’s polynom ial ([Me] Theorem 6.1). It follows that forany Coxeter system $(W, S)$, either $f_{(W,S)}\in \mathcal{T}$or $M(f_{(W,S)})\geq M(f_{(W_{0},S_{0})}$, which solves Lehmer’s problemfor thisclass

ofexamples.

3.3

M-matrices

Let $T\in \mathrm{S}\mathrm{L}(n, \mathbb{Z})$ be amatrixsatisfying

$T=\pm M^{\mathrm{t}\mathrm{r}}M^{-1}$, (3)

where $M\in \mathrm{S}\mathrm{L}(\mathrm{n}, \mathbb{Z})$, and $M^{\mathrm{t}\mathrm{r}}$

isthe transposeof$M$. Then$T^{-1}$ is conjugateto $\pm T^{\mathrm{t}\mathrm{r}}$and hence _$T$ has reciprocal

or

anti-reciprocal characteristic polynomial. Such matrices

are

calledM-matrices.

Howlett [How] showed that if$T$ isthejCoxeterelement of

a

simply-laced Coxeter system

asso-ciatedto

a

graph$\Gamma$, then the Coxeterelement can be writtenin terms of the adjacency matrix $A$

for $\Gamma$

.

Let $A^{+}$ be its upper triangular part. Th

en

setting$M=I-A^{+}$ ,

we

have $T=-M^{\mathrm{t}\mathrm{r}}M^{-1}$.

This gives another proof that the characteristic polynomial of a Coxeter element isreciprocal in the simply-laced case.

Anotherwell-known case is the Alexander matrix ofaknot $(S^{3}, K)$ (see [Rolf] for definitions).

Let $V$ be the Seifert matrix for $(S^{3}, K)$. Then the Alexander polynomialfor $(S^{3}, K)$ is given by

$\Delta_{(S^{3},K)}(t)=|\det(tV-V^{\mathrm{t}\mathrm{r}})|$, up to multiplesof$t^{\pm 1}$

.

If_$V$isinvertible, it follows that the

characteris-ticpolynomial$\Delta(S^{3},K)$ is reciprocal andhas

a

monicrepresentative. Furtherm ore, $\Delta(S^{3},K)(1)=\pm 1$.

Conversely, if $J^{\cdot}$is areciprocal polynomialwith$f\langle 1$) $=\pm 1$, thenthere is aknot $(S^{3}, K)_{\mathrm{t}}$ such that

3.4

Growth rates of

automatic

groups

Let $G$ be afinitely presented group, with generating set $S$, such that $S$ is closed under inverses.

The growth seriesof$G$ is the formal

sum

$\Psi_{(G,S)}(x)=\sum_{\mathrm{i}=0}^{\infty}a_{n}x^{n}$

where$a_{n}$isthenumber of words in$G$ofminimalword length$n$inthe generating set

$S$. The growth

rateof$G$ isgiven by

$\lambda(G, S)=\lim\sup|a_{n}|^{\frac{1}{n}}$

.

(See $[\mathrm{E}\mathrm{C}\mathrm{H}^{+}]$ for more details.) The growth series $\Phi(G,S)$ is rational, for example, when $G$ is

hyperbolic, automatic,

or

a Coxeter group (see forexample $[\mathrm{E}\mathrm{C}\mathrm{H}^{+}]$, [Can], for more details). In

theautomaticcase,

we can

realize$\lambda(G, S)$ as$\lambda(f(G,S))$where$\mathrm{f}(\mathrm{a},\mathrm{s})$is thecharacteristicpolynomial

ofan associated matrix.

Lehmer’s polynomial appears among these examples as follows. Let $(G , Sp1,\ldots,Pk\mathcal{P}1,. ,\mathrm{P}k)$ be

the Coxeter group of reflections through sides of a polygon in the hyperbolic plane with angles $\frac{\pi}{p_{1}}$,$\ldots$,$\frac{\pi}{p_{k}}$, where

$\frac{1}{p_{1}}+\cdots+\frac{1}{p_{k}}<k-2$,

Then Cannonand others [F-P], [C-W], [Floy] calculatethe denominators ofthegrowthseries, and show that $\lambda(G_{p1},.,\mathrm{P}k , S_{p1}, 1p\lambda )$ is aSalem number. In particular, Lehmer’s polynomal $f_{L}$

occurs

as

thedenominatorfor $(G_{2,3,7}, \mathrm{S}_{2,3,7})$ and corresponds to the angle set givingriseto thesmallest area

hyperbolic polygon.

There is

a

close relation between the

automatic

group structure of $(G_{\mathcal{P}1}, .,Pk ,\mathrm{P}1,. .,pk S)$ and the Coxeter element of $(\mathcal{G}_{\rho_{1}},.,p_{k}’ S_{p_{1},..,pk}))$ where $(\mathcal{G}_{\mathrm{p}1{}_{\ddagger}Pk}..,, \mathrm{S}_{\mathcal{P}1,\ldots,\mathrm{P}\mathrm{L}}.)$ is the simply-laced Coxeter

system

associated

to the “star-like” graph with $\mathrm{f}\mathrm{c}$-branches emanating from a central vertex of lengths $p_{11}$

.

._’$Pk$

.

(See, for example, [Hir2]. Although the star-like graphs do not directly

de-fine theauotomatic structures of $(G_{p1},.,\rho_{k}’ S_{\rho 1,.,T^{J}k})$, calculationsin [Hirl] showthat the sequence

$a_{n}$ for the growth series

$\Psi(G_{p_{1}}, ,\mathrm{p}_{k)}^{S}\mathrm{p}_{1}.,p_{k})$

can

be computed directly from the Coxeter element $(\mathcal{G}_{p_{1,\}}\mathrm{p}k}., \mathrm{S}_{\mathcal{P}1\}}.,p_{k})$.

3.5

Dilatations

of pseudo-Anosov

maps

Let $F$ be a compact

orientable

surface with negative Euler characteristic, and let

$\phi$ : $Farrow F$

be ahomeomorphism. The

Thurston-Nielsen

theory [Thu] [FLP] [CB] states that for any surface

homeomorphism $\phi$: $Farrow F$, $\phi$ is isotopic to

some

0 satisfying one ofthe following:

(i) $\Phi$ isperiodic, i.e., $\Phi^{n}$ is theidentity;

(ii) (X) _is irreducible, i.e.,there is a closed

curve

on $F$ invariant under 4 such that complementary

componentshavenegative Euler characteristic;

or

(iii) (X) _is pseudo-Anosov, i.e., there is

a

number $\lambda>1$ and a pair $F^{\pm}$ of transverse

measured

foliations such that

$\Phi(\mathcal{F}^{\pm}\}=\lambda^{\pm 1}F^{\pm}$.

Inthepseudo-Anosovcase, $\Phi$isthe uniqueelementintheisotopyclass of

$\phi$with smallest topological

Suppose $\Phi$ is pseudo-Anosov. Then there is anembedded graph

$\mathcal{G}$ in$F$ representing the spine

of$F$, such that the transitionmatrix $M_{\Phi}$for$\Phi$ restrictedto$\mathcal{G}$ is aPerron-Frobeniusmatrix. Thus,

the characteristic polynomial $f\phi$ of $M_{\Phi}$ is aPerron polynomial. Furthermore, the Perron number

associatedto $f_{\phi}$ equals thedilatation $\lambda(\phi)$

.

The transition matrix

$M_{\Phi}$ has the property that $M_{\Phi}^{\mathrm{t}\mathrm{r}}$

1s conjugate to$M_{\Phi}^{-1}$ and hence its characteristic polynomial $f\emptyset$is reciprocal.

$\langle \mathrm{a})$ (b)

Figure1: Braids with smalldilatation.

In [HK] westudytwofamilies of pseudo-Anosov maps

on

a marked disk

associated

to thebraids $\beta_{m,n}(\mathrm{a})$ and $\sigma_{m,n}(\mathrm{b})$ drawninFigure 1, and showthat their dilatations satisfythe characteristic equations

$x^{n+1}R_{m}(x)+(R_{m})_{*}(x)$

for $\beta_{m,n}$ and, when $n\geq m+2$,

$x^{n+1}R_{m}(x)-(R_{m})_{*}(x)$

for$\sigma_{m,n}$, when$\mathrm{e}$

$R_{m}(x)=x^{m}(x-1)-2$.

The polynomials $R_{m}$ have$\tau r\iota$ roots outside the unit circle. Thus, $R_{1}(x)$ isa P-V polynomial, and hence by Theorem 1 thedilatations of$\beta_{1,n}$ and$\sigma_{1,n}(n\geq 3)$ areSalemnumbers and

are

monotone

(decreasingfor$\beta_{1,n}$ andincreasingfor$\sigma_{1,n}$). For all$m$,the polynomials$R_{m}$arePerronpolynomials, and hence by Theorem 3 the dilatations areeventually monotone (again, decreasing for5 $n\iota,r\iota$ and

increasing for$\sigma_{m,n}$). Since $\lambda(R_{m})$ approaches 1

as

$77\mathrm{L}$ goesto infinity, it follows that for any

$\epsilon>0$,

it is possibleto make$m$ and$n$ large enoughso that thedilatations are within$\epsilon$ of 1. The pseudo-Anosovmaps definedby$\beta_{m,n}$ and$\sigma_{m}$,

$n$lift to homeomorphisms ofgenus$g$ compact

surfaces with $b$ boundary

components

via double covering, where, if$m+n$ is even, $g= \frac{m+n}{2}$ and

$b=1$, and if $\prime r\iota$$+n$ isodd, $g= \frac{r\mathrm{n}+n-1}{2}$ and $b=2$ . Let $\phi_{g}$ be the lift of$\sigma_{g-1,g+1\sim}$ Then

$\log(\lambda(\phi_{g}))=\log(\lambda(\sigma_{g-1,g+1}))-\cdot\frac{1}{g}$,

and we

recover

Penner’s result

on

least dilatations ofpseudo-Anosov maps

on

orientabie

genus

$g$

surfaces. The lifts of$\beta_{m,n}$ are themonodromyof fibered two-bridgelinks (seealso [Bri] and [Hir4]). Reflecting $\beta_{m,n}$

across

the axis containing the marked points, we

see

that the two braids are

in the form given in Figure 2. A conjecture of de Carvalho and Hall $[\mathrm{d}\mathrm{C}\mathrm{H}]$, predicts that under

certain conditions

on

$B$ and for $n$ large enough, braids of the form given in Figure 2 (left) will

determine pseudo-Anosov maps whoseinvariant fibrations will havenice limiting behavior. Thus,

it is natural to askthe following.

Question 2 Underwhat conditions on B will the braids in the

_form

given in Figure 2 (left) have

$\zeta \mathrm{a})$ ( $|\begin{array}{l}-|\prime\prime,r_{\prime}.\prime\end{array}|$

1

(b) $‘\ovalbox{\tt\small REJECT}_{1}|$ $\frac{\mathrm{m}}{\mapsto^{1}}||_{\tau}|\ldots..|$

Figure 2: Braid form.

For$\beta_{m,n}$ and$\sigma_{m}$,_$n$the limitingbehavioristhe same, andthis

occurance

explainstheappearance

of$R_{m}$ for the characteristic equationsof both braid families.

Problem 3 Characterize pairs

_of

distinct braids which

if

plugged into B will give rise to

Salem-Boydsequences associated to thesamepolynomial P anddiffering $\dot{\mathit{0}}$y the sign in

front of

P.

3.6

Homological dilatation

Let$\phi$: $Farrow F$be a surface homeomorphlism, andlet

$\phi_{*}$be therestrictionof$\phi$tothe first homology

Hx$(\mathrm{F};\mathbb{R})$

.

If$\Phi$ isapseudo-Anosovrepresentativeof the isotopyclassof$\phi$,thensince

$M_{\Phi}$ measures

thegrowth rate of word lengths of$\pi_{1}(F)$ under iterationsof$\Phi$ (see, for example, [FLP], [BH]) we

have in general

$\lambda(\phi_{*})\leq\lambda(\phi)$.

If, in addition, the invariant foliations $F^{\pm}$ are orientable, then the largest eigenvalue $\lambda(\phi_{*})$ of $\phi_{*}$,

calledthe homological dilatationof$\phi$, equals $\lambda(\phi)[\mathrm{R}\mathrm{y}\mathrm{k}]$.

A link $(S^{3}, K)$ is apair where $K$ is the disjoint unionof afinitenumber ofsmoothly embedded

circles in $S^{3}$. Alink $(S^{3}, K)$ is

fibered

if for

a

regular neighborhood$U(K)$ of$K$ in$S^{3}$, $S^{3}\backslash U(K)$ is

alocally trivial fiberbundle over$S^{1}$. This fiber bundle structure

over

$S^{1}$ isnot necessarily unique when $K$ has

more

than one compoent.

Given a

fibration let

$\Sigma$ be a fiber. Then $s^{3}\backslash U(K)$ is

homeomorphicto the product of $\Sigma \mathrm{x}$ $[0,1]$ modulo

an

identification $(x, 1)=(\phi(x), 0)$, where $\phi$ is

a

surface homeomorphism from$\Sigma$ to itself, i.e., it is the mapping torus for $\phi$. If$K$ is aknot, then

the

characteristic

polynomial of $\phi_{*}$ equals $\Delta\langle S^{3},K$). In general, if$f$ is

a

monicreciprocal integer

polynomial, then there isa fibered link suchthat $J$ equals thecharacteristic polynomial of

$\phi_{*}$ uP

to amultiple of $(x$ –1$)$ [Kan]. It thus follows that any question about algebraic integers canbe

translated to a question about homologicaldilatations offibered links.

Problem 4 Characterize

fibered

links with small Mahler

measure greater

than

one.

Usingthesimilarforms ofAlexander matrices and Coxeter elements

described

inSection 3.3, it

ispossible to

construct

many examples offibered knots (andlinks) such that

$f_{\phi}.(x)=f_{(W,S)}(-x)$,

One suchexample isthe (-2,3,7)-pretzel knot $K_{2,3,7}$,which is afiberedknot associated to the $E_{10}$ Coxeter system. Since $K_{2,3,7}$ has onlyone component,

$f_{\phi}.(x)=\Delta_{K_{2,3,7}}(x)$.

Thus, A$K_{2,3.7}$$(-x)$ $=f_{L}(x)$ is the Lehmer polynomial (cf. Section 3.2).

If $K_{n}$ is obtained from a fibered link $K_{0}$ by plumbing a $(2-n)$-torus link (see, for example, [Mur] and [Har] for definitions), we say that $K_{n}$ is obtained from $K_{0}$ by iterated Hopfplumbing. In [Hir4] , weshow that the characteristic polynomial ofthe homological monodromyof such a $K_{n}$ is a Salem-Boyd sequence. As an example, braids ofthe form given in Figure 2 define via double

covering homeomorphisms ofsurfaces, whichare obtainedby iterated Hopf plumbing.

Problem 5 Let$\phi$: $Farrow F$ beahomeomorphism

of

a

surface

with boundary, and let

$\phi_{n}$ : $F_{n}arrow F_{n}$

be obtained by iterated Hopfplumbing. Under what conditions on $\phi$ and$n$ is $\phi_{n}$ pseudo-Anosov,

andin this case do the dilatations satisfy a Salem-Boyd sequence?

References

[A’C] N. A’Campo. Surles valeurs propres de la transformation de Coxeter. Invent Math. 33

(1976), 61-67,

[BH] M. Bestivina and M. Handel. Traintracks and automorphism soffreegroups. Ann. Math. II. s. 135 (1992), 1-51.

[Boydl] D. W. Boyd. Speculations concerning the range ofMahler’s

measure.

Canad. Math. Bull.

24 (1981), 453-469.

[Boyd2] D.W. Boyd. Small Salem numbers. Duke Math. J. 44 (1977), 315-328.

[Bri] P. Brinkmann. A Note

on

pseudo-Anosov maps with small growth rate. Experimental

Math, ₁₃ (2004), 49-53.

[C-W] J. Cannonand P. Wagreich. Growthfunctions ofsurface groups. Math. Ann. 293 (1992),

239-257.

[CB] A. Casson and S. Bleiler. Automorphisms

of

surfaces after

Nielsen and Thurston. Cam-bridge UniversityPress, 1988.

[dCH] A. de Carvalho and T. Hall. The forcingrelation for horseshoe braid types. Experiment. Math. 11 (2002), 271-288.

[DP] $\mathrm{J}$ DufresnoyandCh,Pisot.

\’Etude

de certainesfonctionsmeromorphes borneesurlecercle

unite, application \‘aun ensemblefermed’entiersalgebriques. $Ann$, Sci,

\’Ecole

Norm. Sup.

72 (1955), 69-72.

$[\mathrm{E}\mathrm{C}\mathrm{H}^{+}]$ D. Epstein, J. Cannon, D. Holt, S.Levy, M. Paterson, andW. Thurston. Wordprocessing

in groups. Jones and BartlettPublishers, Boston, MA, 1992.

[FLP] A. Fathi, F.Laudenbach, and V. Poenaru. Travauxde Thurstonsttr les surfaces, volume

66-67. Societe MathematiquedeFrance, Paris, 1979.

[Floy] W. J. Floyd. Growth of planarCoxeter groups,$\mathrm{P}.\mathrm{V}$.numbers, andSalemnumbers. Math.

[F-P] W. J. Floyd and S. P. Plotnick. Symmetriesof planar growth functions ofCoxeter groups. Invent Math. 93 (1988), 501-543.

[Gan] F. Gantmacher. The Theory

_of

Matrices vol. 1. ChelseaPublishing Co., NewYork, 1959. [G-H] E. Ghate and E. Hironaka. The Geometry of Salem numbers. Bulletin

_of

Arner. Math.

Soc. 38 (2001), 293-314.

[G-Mc] D. Gross and C. McMullen. Automorphisms ofeven unimodular lattices. J. Algebra (to appear).

[Har] J. Harer. How toconstruct allfiberedknots andlinks. Topology 21 (1982), 263-280.

[Hirl] E. Hironaka. The LehrnerPolynom ialand Pretzel Knots. Bulletin

of

CanadianMath. Soc. 44 (2001), 440-451.

[Hir2] E. Hironaka. Coxeterlinks, McKay’scorrespondence and 2,3,7. Contemp. Math. Proc 324

(2003), 123 – 138.

[Hir3] E. Hironaka. Chorddiagrams andCoxeterlinks. J. London Math. Soc.69 (2004),

243-257.

[Hir4] E. Hironaka. Salem-Boyd sequences and Hopf plumbing, preprint (2005).

[HK] E. Hironaka and E. Kin. A family ofpseud$0$-Anosovbraidswithsmalldilatation, preprint

(2005).

[How] R. Howlett. Coxeter groups and$M$-matrices. Bull London Math. Soc. 14 (1982),

137-141.

[Hum] J. Humphreys.

Reflection

groups and Coxeter groups. Cambridge University Press, Cam-bridge, 1990.

[Can] J.W.Cannon. The combinatorial structureof cocompact discrte hyperbolicgroups. Geom. Dedicate 16 (1984).

[Kan] T. Kanenobu. Module d’Alexander des noeuds fibres et polynome deHosokawa des

lace-mentsfibr\’es. Math. $Sem$

.

Notes Kobe Univ. 9 (1981), 75-84.

[Leh] D. H. Lehmer. Factorization ofcertain cyclotomic functions. Ann.

of

Math. 34 (1933),

461-469.

[Mc] C. McMullen. Coxeter

groups,

Salemnumbers and the Hilbertmetric. Publ Math. Inst.

Hautes

\’Etudes

Sci. 95 (2002),

151-183.

[Mos] M. Mossinghoff. Lehmer’s problem, $http.\cdot//www$

.cecm.

$sfu$.$co/mJm/Lehmer$(2003),

[Mur] K. Murasugi. On the genus of the alternatingknot. 1,11. J. Math.Soc. Japan104 (1958),

94-105,235-248.

[Rolf] D. Rolfsen. Knots and Links. Publish orPerish, Inc,Berkeley, 1976.

[Ryk] E. Rykken. Expanding factors for pseudo-Anosov homeomorphisms. Michigan Math. $J$

.

46 (1999),

281-296.

[Sal] R. Salem. A

remarkable

class ofalgebraicintegers.Proofof a conjectureofVijayaraghavan.

[Seif] H. Seifert. $\ddot{\mathrm{U}}$

berdas Geschlecht von Knoten. Math. Ann. 110 (1934), 571-592,

[Sie,] C. L. Siegel. Algebraic integerswhose conjugateslie onthe unit circle. DukeMath. J. 11 (1944), 597-602.

[Smy] C. J. Smyth. On the product of the conjugates outside the unit circle of an algebraic

integer. Bull. London Math. Soc. 3 (1971), 169-175.

[Thu] W.Thurston. Onthegeometry anddynamicsofdiffeomorphisms ofsurfaces. Bull. Amer. Math. Soc. (N.$S.$) $19$ (1988), 417-431