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\dateposted{May 28, 1999}
\PII{S 1079-6762(99)00062-1}






%\let \c@equation=\c@theorem



\title{The Hilbert--Smith conjecture for quasiconformal actions}

\author{Gaven J. Martin}

\address{Department of Mathematics,
The University of Auckland,
Private Bag 92019,
New Zealand}


\copyrightinfo{1999}{American Mathematical Society}

\subjclass{Primary 26A24, 30C60, 53A04, 54F65}
\date{November 9, 1998}

\commby{Walter Neumann}


\thanks{Research supported in part by a grant from the N.Z. Marsden Fund.}

This note announces a proof of the Hilbert--Smith conjecture in the
quasiconformal case:  A locally compact group  $G$  of quasiconformal 
homeomorphisms acting effectively on a Riemannian manifold is a Lie group.
The result established is true in somewhat more generality.


This note announces a proof of the Hilbert--Smith conjecture for
quasiconformal actions on
Riemannian manifolds and related spaces.

We were led to this conjecture through our study of two
important areas of geometric function theory.  The first concerns the
problem of unique analytic
continuation of solutions to general Beltrami systems in higher dimensions:
\[ Df^t(x) G(f(x)) Df(x) = J_f(x)^{2/n}H(x) \hskip10pt \textup{a.e.,}\]
where $G$ and $H$ are bounded measurable functions valued in the space of
symmetric positive definite
$n\times n$ matrices of determinant $1$ ($Df$ is the Jacobian matrix of $f$,
and $J_f$ its determinant). Solutions to
these nonlinear systems of PDE's  are {\em quasiregular} or {\em
quasiconformal} mappings.  These mappings
play an important role in many aspects of modern geometry and analysis.
For instance, in nonlinear elasticity,
topological index theory,  harmonic analysis,  nonlinear potential theory,
and so forth.  These mappings are natural
higher dimensional analogues of  analytic or conformal mappings in
$\IC$.  See
\cite{Ri} for a good starting point to the theory.  The second problem we
were concerned with was the
classification of the dynamics of mappings which are rational with respect
to a bounded measurable Riemannian
stucture, now called {\em uniformly quasiregular} mappings.  This theory
was initiated in
\cite{IM}, but see
\cite{May} too.  Largely because of Rickman's version of Montel's normal
criterion,  this theory closely parallels the classical iteration theory of
rational endomorphisms of
$\oC$,  but again in higher dimensions.  We were interested in proving
Siegel's theorem concerning the local
conjugacy of rotational dynamics about nonattracting fixed points in the
Fatou set.

How these problems are related and how their affirmative solution follows
from the Hilbert--Smith
conjecture is discussed in \cite{M}, where more details can be found for the
sketch of proof of our main
results given here.

Hilbert's fifth problem \cite{Hi} asks if every finite dimensional locally
Euclidean topological group
is necessarily a Lie group.  This problem was solved by von Neumann in 1933
for compact groups and by
Gleason and Montgomery and Zippin in 1952 for locally compact groups;  see
\cite{MZ} and the
references therein.

A more general version of the fifth problem asserts that among all locally
compact groups $G$  only Lie
groups can act effectively on finite dimensional manifolds.  This problem
has come to be called the
Hilbert--Smith Conjecture.  It follows from the work of Newman and of Smith
together with the
structure theory of infinite abelian groups that the conjecture is
equivalent to the special case
when the group $G$ is isomorphic to the $p$-adic integers  $A_p$.

In 1943 Bochner and Montgomery \cite{BM} solved this problem for actions by
Although there is considerable literature on this problem, for our purposes
the fundamental result
we need was established by Yang \cite{Y} in 1960.

\begin{theorem}  If  G  is a $p$-adic group acting effectively on a homology
$n$-manifold  $M$,  then the orbit space  $M/G$  is of homological dimension

Yang's result is based on the construction of certain exact sequences in a
modified version of
Smith homology theory and might nowadays be regarded as standard.  There
are relatively nice $n$-dimensional
metric spaces on which the $p$-adics can act increasing the dimension

It took until 1997 for Yang's result to be used in an effective way.
Repov\v{s} and \v{S}\v{c}epin
used it to prove the Hilbert--Smith Conjecture for Lipschitz actions by
comparing the Hausdorff and
cohomological dimensions of the orbit space,
\cite{RS}.  R.~D.~Edwards has announced work toward a solution of the
Hilbert--Smith conjecture for free

Once we became aware of the work \cite{RS}, it became clear how one might
establish the result in the more general
setting of quasiconformal mappings.  There are technical difficulties
because quasiconformal mappings are not
Lipschitz and although they enjoy H\"older continuity properties, these are
not sufficient to directly apply the
method of Repov\v{s} and
\v{S}\v{c}epin.  We must thank K. Astala for pointing out how to make an
important step in our argument.  Our main
results are as follows.  We give relevant definitions and a sketch of proof
in the next sections.

\begin{theorem}  Let $G$ be a locally compact group acting effectively and
quasiconformally on a
Riemannian manifold.  Then $G$ is a Lie group.

As noted above,  this result is implied by the following

\begin{theorem} The group $A_p$ of $p$-adic integers cannot act effectively by
quasiconformal homeomorphisms on any Riemannian manifold.

\begin{remarkn}  Sullivan has shown that every topological
$n$-manifold ($n\neq 4$) admits a
unique quasiconformal structure, \cite{Su}. (The situation in
$4$ dimensions is more complicated;
see \cite{DS}.)  Thus it is possible to speak of quasiconformal actions on
an arbitrary topological
$n$-manifold,  $n\neq 4$.  It is in this setting that our results hold and
thus the hypothesis
that the manifold is Riemannian is largely unnecessary.  However, if  $G$ is
a compact group acting
effectively on a topological manifold,  it is far from clear that the
action can be made
quasiconformal (indeed this would solve the general Hilbert--Smith
conjecture ($n\neq 4$) in view of
our result).
\section{Quasiconformal mappings}

Let $\Omega$ and $\Omega^\prime$ be open subsets of $\oR^n$, and  let
$f:\Omega \to \Omega^\prime$ be a
homeomorphism. Quasiconformal mappings are first and foremost mappings of
bounded distortion.
The infinitesimal distortion of  $f$ at  $x$ is
H_f(x) = \limsup_{r\to 0} \frac{\max_{|h|=r} |f(x+h)-f(x)|}{\min_{|h|=r}

\begin{definitionn}  A homeomorphism $f:\Omega\to\Omega^\prime$
between subdomains of $\IR^n$
is {\em quasiconformal} if there
$H<\infty$ such that
H_f(x) \leq H \quad\mbox{ for all $x\in \Omega$}.
The essential supremum of the function $H_f(x)$ is called the maximal
distortion or dilatation of $f$.
Unfortunately, this definition,  while aesthetically pleasing, is difficult
to work with. Also it is not lower
semicontinuous with repect to limits,  a property we shall need.  It is
more usual to study quasiconformal mappings
as Sobolev functions satisfying the differential inequality between the
Jacobian matrix
$Df$ and its determinant $J_f$,
|Df(x)|^n \leq K J_f(x)\quad \textup{ a.e. in } \Omega.
We say that a $W^{1,n}_{loc}(\Omega,\IR^n)$ function satisfying this
inequality $f$ is $K$-quasicon\-for\-mal.
There are other natural definitions involving distortion of moduli (or
other conformal invariants), and an
important part of the foundations of the theory is the equivalence of these
definitions.  The generalisations to
quasiconformal mappings between quasiconformal manifolds and so forth are
straightforward (see
\cite{Ri}).  It is also clear that Lipschitz and smooth maps are locally
quasiconformal and therefore
quasiconformal on relatively compact subdomains.

A topological transformation group  $G$  acting
on a quasiconformal $n$-manifold $M$
is said to act quasiconformally if each $g\in G$ is quasiconformal.

It is important to note that we have made no {\em a
priori} assumption of a uniform bound on
the dilatation of each element of $G$.
\section{Sketch of proof for Theorem 1.3}

We break the proof into a number of steps.

It is enough to consider the $p$-adics acting effectively and
as a transformation group $G$ of a domain $\Omega$ in $\IR^n$ (the ideas
are the same).
For a positive integer $n$ we set
E_n=\{g\in G: K_g \leq n \}.
Then $E_n$ is a countable family of closed sets whose union is $G$.  By the
Baire category theorem there is some
$E_n$ with nonempty interior. Translating by an element of $G$ we observe
that there is some $m\geq 0$ with $E_m$
containing a neighbourhood of the identity.  Any neighbourhood of the
identity in the
$p$-adic group contains an isomorphic copy of the $p$-adics and so, after
replacing $G$
with this copy, we may assume that there is $K<\infty$ such that each
$g\in G$ is $K$-quasiconformal.  That is,  $G$  is a uniformly
quasiconformal group.

$G$ has an invariant Haar measure $dg$ of total mass $1$.  Point
evaluation $x\mapsto g(x)$ is
continuous on $G$ and thus we may construct an invariant metric on
$\Omega$ as
\[ d_G(x,y) = \int_G |g(x)-g(y)| dg. \]
This metric generates the usual topology of $\Omega$.  In fact, since each
$g\in G$ is now locally $1/K$-H\"older
continuous,   the invariant metric is locally H\"older equivalent to our
background metric.  We want to estimate
the Hausdorff dimension of the metric space $(\Omega,d_G)$.  This is a
local problem.  The compactness of $G$
allows us to assume $Vol(\Omega)<\infty$.  Let
$r$ be a small number and cover a relatively compact open subset $U$ of
$\Omega$ by a family ${\mathcal F}$ of balls of
$r$ in the background metric.  We may refine this cover using the
Besicovitch covering theorem \cite{Maz} so as to
have bounded overlap (independent of
$r$).  That is, there is a fixed constant
$c$ such that each $B\in {\mathcal F}$  meets at most $c$ other elements of
${\mathcal F}$.  We want to look at the sum
\[ \sum_{B\in {\mathcal F}} d_G(B)^n, \]
where $d_G(B)$ is the diameter of $B$ in the invariant metric,  and bound
it by a number independent of $r$.  This
will imply that the Hausdorff dimension of the space
$(\Omega,d_G)$ is at most $n$.  For each $B\in {\mathcal F}$ let $x_B$ denote
its center and $y_B$  a point
on the boundary with $d_G(B) \leq 2 d_G(x_B,y_B)$.  Then
\lefteqn{\sum_{B\in {\mathcal F}} d_G(B)^n }\\ & \leq &   C_n \; \sum_{B\in
{\mathcal F}} \left( \int_G
|g(x_B)-g(y_B)|dg \right)^n
\\ &\leq &  C_n \; \sum_{B\in {\mathcal F}} \int_G |g(x_B)-g(y_B)|^n dg
\hskip10pt \mbox{(Jensen's inequality)} \\
& \leq  &  C_{n,K} \; \sum_{B\in {\mathcal F}} \int_G
\min_{|h|=r}|g(x_B)-g(x_B+h)|^n dg
\hskip10pt  \mbox{($K$-quasiconformality)} \\
& \leq  &  C_{n,K} \; \sum_{B\in {\mathcal F}} \int_G Vol(g(B)) dg
= C_{n,K} \;  \int_G \sum_{B\in {\mathcal F}} Vol(g(B)) dg \\
& \leq  & C_{n,K} Vol(\Omega)  \hskip10pt \mbox{(Balls have bounded overlap).}
We have used the usual convention that the constant may change from line to
line, but have indicated its
dependence on the parameters.  The quasiconformal fact used is essentially
Mori's theorem and the reduction to the
uniformly quasiconformal case is crucial here.  We have now shown
that $(\Omega,d_G)$ has Hausdorff dimension $n$.

As the metric $d_G$ is invariant,  the orbit space $\Omega/G$
inherits the metric
$\rho(\eta,\zeta) = d_G(G(x),G(y))$, where $\eta = \pi(x)$ and $\pi:\Omega
\to \Omega/G$ is the projection.  The
map $\pi$ is evidently a contraction.  Hausdorff dimension cannot increase
under a contraction mapping.  Therefore
the Hausdorff dimension of $\Omega/G$ is at most $n$.  We now use a few
simple facts from dimension theory.  First,
the Hausdorff dimension dominates the topological dimension,  which in turn
then dominates the cohomological
dimension.  The result now follows as we have obtained a contradiction to
Yang's theorem.


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$4$-manifolds},  Acta Math. \textbf{
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G\"ottingen (1900),
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Ann. Acad. Sci. Fenn. Math. \textbf{21}
(1996), 241--254.
\bibitem{M} G. J. Martin, {\em UQR mappings,  Siegel's theorem and the
Hilbert--Smith conjecture},
in preparation.
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conjecture for actions by
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\bibitem{Ri}  S. Rickman,  {\em Quasiregular mappings},  Springer-Verlag,
\bibitem{Su}  D. Sullivan,  {\em Quasiconformal and Lipschitz structures},
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