Geometry, Topology and Dynamics Seminar
Peter Shalen
UIC
(non)-Units in algebraic number fields, and Margulis numbers of hyperbolic 3-manifolds
Abstract: If $P$ is a point of a hyperbolic manifold $M$, and if $x$ is an element
of $\pi_1(M,P)$, let $L(x)$ denote the minimum length of a loop based at $P$
and representing $x$. If $x$ and $y$ are elements of $\pi_1(M,P)$,
set $M(x,y)=\max(L(x),L(y))$. We define the Margulis number $\mu(M)$ to be
the infimum of $M(x,y)$ as $P$ ranges over all points of $M$ and $(x,y)$
ranges over all pairs of elements of $\pi_1(M,P)$ such that the subgroup
generated by $x$ and $y$ is not virtually nilpotent. It is
perhaps the most fundamental fact in the geometry of hyperbolic
manifolds that $\mu(M)$ is bounded below by a strictly positive
number, the $n$-dimensional Margulis constant, as $M$ ranges over all
hyperbolic $n$-manifolds. For $n = 3$ the largest known lower bound for the
Margulis constant, due to Meyerhoff, is about 0.104. For the case of a
hyperbolic 3-manifold $M$ for which $\pi_1(M)$ has no 2-generator subgroup
of finite index, work by Culler and S, combined with the
Agol-Calegari-Gabai tameness theorem, shows that $\log 3 = 1.09...$ is a
lower bound for the Margulis number of $M$.
In this talk I will describe recent work in which I relate the problem
of estimating the Margulis number to some questions in algebraic
number theory. Let $M$ be a closed, orientable hyperbolic 3-manifold,
which for simplicity I will assume is a mod-2 homology sphere. Then
$\pi_1(M)$ may be canonically identified (up to conjugacy) with a
subgroup $\Gamma$ of $SL(2,\mathbb{C})$, and the traces of elements of $\Gamma$
generate a finite extension $K$ of $\mathbb{Q}$. If $M$ is not a Haken manifold,
then it has "integral type" in the sense that the traces of the
elements of $\Gamma$ are elements of $O_K$, the ring of integers in $K$.
Theorem: Let $M$ be a closed, orientable hyperbolic 3-manifold. Suppose
that $M$ is a mod-$p$ homology sphere for $p$ = 2, 3, and 7, and is of
integral type. Let $P$ be a point of $M$ and let $x_1$ and $x_2$ be
non-commuting elements of $\pi_1(M,P)$, and let $\sigma_i$ denote the trace
of $x_i$ (identified with an element of $\Gamma\subset SL(2,\mathbb{C})$).
(1) If $\sigma_i$ and $\sigma_i^2 - 2$ are non-units in $O_K$ for $i = 1$ and
for $i = 2$, then $M(x_1,x_2) > 0.39$.
(2) If $\sigma_i + 1$ and $\sigma_i - 1$ are non-units in $O_K$ for $i = 1$ and
for $i = 2$, then $M(x_1,x_2) > 0.3$.
I have applied this result to estimate Margulis numbers in the case
where $K$ has degree 2 or 3 over $\mathbb{Q}$. Let us retain the assumptions that $M$
is a closed, orientable hyperbolic 3-manifold, is a mod-$p$ homology
sphere for $p$ = 2, 3, and 7, and is of integral type. If $K$ is a
quadratic field I have shown that $\mu(M) > 0.39$. The case where $K$ is a
cubic field is not yet completely written up, but as this abstract
goes to press it appears that I can show that $\mu(M) > 0.3$ in this
case.
Monday September 15, 2008 at 3:00 PM in SEO 612