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\rightline{to appear in \it Math. Proc. Camb. Phil. Soc.}
\bigskip
\centerline{\bf Spotting Infinite Groups}
\bigskip
\centerline{\sc By Daniel Allcock\footnote\dag{\rm Supported by NSF Graduate
and Postdoctoral Fellowships.}}
\medskip
\centerline{\it Department of Mathematics, University
of Utah, Salt Lake City, UT \rm 84102}
\centerline{\it allcock@math.utah.edu}
\medskip
\centerline{({\it Received $6$ December $1996$; revised $2$ May
$1997$})}
\bigskip
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%\title{Spotting Infinite Groups}
\centerline{\it Abstract}
\smallskip %\abstract
We generalize a theorem of R. Thomas, which sometimes allows one to tell
by inspection that a finitely presented group $G$ is
infinite. Groups to which his theorem applies have
presentations with not too many more relators than generators,
with at least some of the relators being proper powers. Our
generalization provides lower bounds for the ranks of the
abelianizations of certain normal subgroups of $G$ in terms of
their indices. We derive Thomas's
theorem as a special case.
\section{1}{Introduction}
There is a very simple theorem which states that any group
with a presentation with at least one more generator
than relations is infinite.
To see this, one simply abelianizes the group and uses linear algebra to
conclude that the abelianization is infinite. It is also classical that the
$(p,q,r)$ triangle group, given by the presentation
$$
\langle a,b\,|\, 1 = a^p = b^q = (ab)^r \rangle ,
$$
is infinite when $1/p + 1/q + 1/r \leq 1$.
One sees this
by realizing the triangle group as a group of transformations of the
Euclidean plane or the hyperbolic plane,
and simply observing that the group is infinite. (See
\cite{cox:gens-and-rlns}.)
It is remarkable that these two facts stem from the same source:
In \cite{thomas:infgps}, R. Thomas showed that when a certain
technical condition applies,
a relator which
is a $p$th power only counts as ``$1/p$ of a relator''
for the purpose of
comparing the number of generators with the number of relations.
His theorem sometimes allows one to conclude ``at a glance''
that a group is infinite.
We will state
his result precisely, as a corollary of our main
theorem.
We generalize Thomas's theorem by
obtaining information about the ranks of the abelianizations of
certain subgroups of a group $G$, if $G$ has a presentation with
at least one more
generator than relators (when relators which are powers
are counted as ``fractions of relators''). The proof is simple and
geometric; our principal tool is elementary homology theory.
In \cite{bergman:infgps} G. Bergman has obtained a further
generalization of Thomas's theorem, which allows one to obtain
results similar to ours for groups with somewhat more complicated
presentations than the ones we treat. His techniques are
algebraic and considerably more sophisticated than ours. He also
surveys related results.
This paper is derived from part of the author's
Ph.D. dissertation
\cite{dja:thesis}.
\section{2}{Theorem and Proof}
{\sc Theorem.}
{\it Let $G$ be a group with presentation
$$
\langle a_1 \ldots a_n \,|\, 1 = w_1^{r_1} = \cdots = w_m^{r_m} \rangle
$$
where each $w_j$ is a word in the $a_i$ and their
inverses. Suppose that $H$ is a normal
subgroup of $G$ of index $N<\infty$ and
that for each $j$, $w_j^k\not\in H$ for $k=1,\dots,r_j-1$.
Then the rank of the abelianization of $H$ is at least
$$
1+N\left(n-1-\sum_i{1\over r_i}\right).
$$}
\beginproof{Proof.}
We begin by building the
2-dimensional CW complex $K$ associated to the
presentation. $K$ has one 0-cell,
$n$ 1-cells, and $m$ 2-cells; the 1-cells are identified with the generators
$a_i$ of $G$, and the 2-cells are attached according to the relators
$w_j^{r_j}$. The fundamental group of $K$ is isomorphic to $G$.
The Euler characteristic $\chi (K)$ is equal to $1-n+m$, and so
the covering space $\tilde{K}$ associated to the subgroup $H$ of $G$
has Euler characteristic
$\chi (\tilde{K}) = N(1-n+m)$. Since $\tilde{K}$ is
connected and only has cells in dimensions $\leq2$,
$H_0(\tilde{K}) \cong \Z$ and $H_i(\tilde{K}) = 0$ for $k>2$.
The abelianization of $H$ is isomorphic
to $H_1(\Ktil)$; we
will estimate the rank of $H_2(\tilde{K})$ and this will give us
the needed information about the rank of $H_1(\tilde{K})$.
Take a close look at $\tilde{K}$. Its $1$-skeleton is a copy of the Cayley
graph $\G$ of $G/H$
with respect to the generating set $\{a_1,\ldots,a_n\}$.
Attached to it are $Nm$ 2-cells, since there are $N$
lifts of each 2-cell of $K$. For each vertex $v$ of $\G$ and for
each $j$ there
is a 2-cell of $\Ktil$ whose boundary is the closed edge-path
beginning at $v$ and tracing out the path $w_j^{r_j}$.
Fix $j$ and observe that the 2-cells associated to the
vertices $v\cdot w_j^k$ (as $k$ varies from $1$ through
$r_j$) all have the same
boundary. Furthermore, since
$w_j^k\not\in H$ for each
$k=1,\dots,r_j$, these are distinct vertices and so we obtain
$r_j$ disks all sharing a common boundary.
There is one such set of $r_j$ disks for every set $\{v\cdot
w_j^k\,|\,k=1,\dots,r_j\}$, so there are a total of $N/r_j$ such sets.
We may form a new
complex $L$ from $\tilde{K}$ by removing $r_j-1$
two-cells from each such set of $r_j$, for each $j=1,\ldots,m$.
We can rebuild
$\tilde{K}$ from $L$ by simply replacing the
removed 2-cells. Since the boundary of each adjoined disk is
already a boundary in $L$, a simple Mayer-Vietoris argument shows that
adjoining any of these 2-cells to $L$
increases the rank of the second homology of the complex. Therefore
rank($H_2(\tilde{K})$) is at least equal to the number of 2-cells added
to $L$ to obtain $\tilde{K}$. For each $j$, there are $N/r_j$
sets of disks, and so after summing over $j$ we obtain
$$
\hbox{rank}\,(H_2(\tilde{K})) \geq \sum_{j=1}^{m}
{N\over r_j}(r_j-1).
$$
Since
$$
\chi(\Ktil)=N(1-n+m)=\hbox{rank }H_0(\Ktil)
-\hbox{rank }H_1(\Ktil)+\hbox{rank }H_2(\Ktil),
$$
we see that
$$
N(1-n+m)\geq 1-\hbox{rank }H_1(\Ktil)+\sum_{j=1}^m N(1-1/r_j).
$$
Rearranging terms, we obtain the claimed result.
\endproof
{\sc Corollary
\rm \cite{thomas:infgps}.}
{\it Let $G$ be a group with presentation
$$
\langle a_1 \ldots a_n \,|\, 1 = w_1^{r_1} = \cdots = w_m^{r_m} \rangle
$$
where each of the $w_j$ is a word in the $a_i$ and their inverses,
and where the order in $G$
of each $w_j$ is actually $r_j$ (i.e., no collapsing takes place).
Then $G$ is infinite if
$$
n \geq 1 + {1\over r_1} + \cdots + {1\over r_m}.
$$}
\beginproof{Proof.}
If $G$ were finite then the trivial subgroup would have finite
index. The theorem would imply that this subgroup had
abelianization of positive rank, which is absurd.
\endproof
\section{3}{Examples}
We can use the theorem to study the groups
$(p,q,r)$ for $1/p+1/q+1/r\leq1$. A
presentation for this group $G$ is
$$
\langle a,b \,|\, 1 = a^p = b^q = (ab)^r \rangle .
$$
The usual proof that $|G|=\infty$
involves realizing $(p,q,r)$ as a group of
isometries of either the
Euclidean plane or hyperbolic plane.
We proceed instead by exhibiting a representation of $(p,q,r)$
to a finite group such that the images of $a$, $b$ and $ab$ have
orders
$p$, $q$ and $r$, respectively.
The corollary then yields the order of $G$, and we can use the
theorem to estimate the ranks of the abelianizations of
subgroups of $G$.
Let $F$ be a finite field
containing primitive $2p$th, $2q$th and $2r$th roots of
unity. (One can construct such a field by taking a prime $\ell$
not dividing $2p$, $2q$ or $2r$, and then taking $F$ to be the
field of order $\ell^n$ where $\ell^n$ is congruent to $1$
modulo each of $2p$, $2q$ and $2r$.) Let $\a$, $\b$ and $\c$ be
such roots. Let $A$ and $B$ be the matrices
$$
A=\pmatrix{\a & 1 \cr 0 & \a^{-1}\cr}
\quad\hbox{and}\quad
B=\pmatrix{\b & 0 \cr x & \b^{-1}\cr},
$$
where we will choose $x$ later. Since the roots of the
characteristic equations for $A$ and $B$ lie in $F$ and are distinct, the
matrices are diagonalizable and have orders $2p$ and $2q$.
One checks that the characteristic polynomial of $AB$ is
(as a polynomial in the symbol $\lambda$)
$$
\lambda^2+\lambda(-\a\b-\a^{-1}\b^{-1}) +1 -\lambda x.
$$
We may obviously choose $x$ so that $\c$ is a root of this
polynomial, so $AB$ has an eigenvalue $\c$ and thus
is conjugate to an upper-triangular matrix $M$.
Since $AB$ has determinant $1$, the diagonal entries of $M$ are $\c$ and
$\c^{-1}$. Since these are distinct and lie in $F$,
$AB$ is diagonalizable and
has order $2r$. Working modulo $\{\pm I\}$, we find that the
images of $A$, $B$, and $AB$
in $PSL_2(F)$
have orders $p$, $q$, and $r$. Therefore the map $a\mapsto A$
and $b\mapsto B$ defines a homomorphism from $G$ to
$PSL_2(F)$.
We also see that
$a$, $b$ and $ab$ have orders $p$, $q$ and $r$ in $G$, and by the
corollary we conclude that $(p,q,r)$ is infinite.
Now we use our
theorem to estimate the ranks of abelianizations of subgroups of
the kernel $K$ of this representation. If $H\sset K$ is a normal
subgroup of $G$ then
in the ``Euclidean'' case ($1/p+1/q+1/r=1$), we find that
the abelianization of $H$ has rank at least $1$, and
that in the ``hyperbolic'' case ($1/p+1/q+1/r<1$), the estimate of
the rank grows linearly with the index of the subgroup.
One can show that in the Euclidean case, $K\cong\Z^2$ and so
actually has rank $2$; in the hyperbolic case, $K$ is isomorphic
to the fundamental group of a surface of genus $>1$ and so its
proper subgroups are fundamental groups of surfaces of still
larger genus, and so have larger abelianizations. Our theorem
provides a purely algebraic approach to this topological fact.
For $n\geq3$, the presentation
$\langle a,b\,|\,1=a^n=a^{n+1}=b^n=b^{n+1}\rangle$
of the trivial group shows that the no-collapsing hypothesis of
the corollary is essential.
In \cite{thomas:infgps}, Thomas gives a
much more subtle presentation of the trivial group that also
shows the need for the the no-collapsing hypothesis.
\bigskip
\centerline{\sc REFERENCES}
\medskip
\parindent=0pt
\hangindent=3em
\hangafter=1
\cite{dja:thesis}
{\sc D.~J. Allcock.}
{The {L}eech Lattice and Hyperbolic Geometry}.
Ph.D. thesis. U.C. Berkeley (1996).
\hangindent=3em
\hangafter=1
\cite{bergman:infgps}
{\sc G.~Bergman.}
Infinite groups arising from partial presentations of finite groups.
Preprint (1996).
\hangindent=3em
\hangafter=1
\cite{cox:gens-and-rlns}
{\sc H.~S.~M. Coxeter {\rm and} W.~O.~J. Moser.}
{\it Generators and Relations for Discrete Groups.}
Springer-Verlag, 1984.
\hangindent=3em
\hangafter=1
\cite{thomas:infgps}
{\sc R.~Thomas.}
{C}ayley graphs and group presentations.
{\it Proc. Camb. Phil. Soc.} {\bf 103}(1988), 385--7.
\bye