数理解析研究所講究録 1270 巻 2002 年 145-152

145

Uniform discreteness and Heisenberg screw motions
JIANG Yueping

Department of Applied Mathematics,
Hunan University, Changsha 410082, PR China
ypjiang@mail.hunu.edu.cn

John R. PARKER
Department of Mathematical Sciences,
University of Durham, Durham DH13LE, England
J.R.Parker@durham.ac.uk

February 5, 2002
Abstract
We give aversion of Shimizu’s lemma for groups of complex hyperbolic isometries one of whose generators is aHeisenberg screw motion.
Our main result interpolates between known results for groups with agenerator that is avertical translation or aHeisenberg rotation. We
also give an interpretation of our result in terms of the relation between radii of isometric spheres and their distance from the axis of theHeisenberg screw motion.

1Introduction
Shimizu’s lemma [12] gives anecessary condition for asubgroup of PSL(2, R)
containing aparabolic element fixing
to be discrete. It was generalisedfor discrete groups of higher dimensional real hyperbolic isometries contain$\infty$

ing aparabolic element by Leutbecher [8], Wielenberg [14], Ohtake [9] andWaterman [13]. The hyperbolic plane is not only real hyperbolic 2-space, but also complex hyperbolic 1-space. Therefore it is natural togeneralise Shimizu’s lemma to discrete groups of isometries of higher dimensional complex hyperbolic
space containing aparabolic element. Thisis part of awider project to give generalisations of Jorgensen’s inequalit$\mathrm{H}_{\mathbb{R}}^{2}$$\mathrm{H}_{\mathbb{C}}^{1}$$\mathrm{H}_{\mathbb{C}}^{n}$146
are eifor complex hyperbolic space, see $[1, 4]$ . Parabolic isometries ofther Heisenberg translations or screw-parabolic transformations (also calledelliptic axabohc transformations). The latter are the composition of an elliptic transformation and aHeisenberg translation preserving the fixed point
set of the elliptic map. For screw-parabolic maps in (complex) dimension$n=2$ the Heisenberg translation involved is necessarily vertical, although
in higher dimensions non vertical translations are also possible. The elliptictransformation and its fixed point set are called the holonomy and the axisof the screw parabolic transformation respectively.
In [6], Kamiya generalised Shimizu’s lemma for vertical translations andin [7] he gave ageometric version which says that there must be aprecisely
invariant horoball whose height depends only on the length of the translation. In [10], Parker observed that Kamiya’s result may be extended togroups containing ascrew-parabolic map whose holonomy has finite order.
On the other hand, also in [10], he showed that, for groups containing a
non-vertical Heisenberg translation (Proposition 7.3) or ascrew-parabolic
map with infinite order holonomy (Proposition 6.4), there is no precisely invariant horoball (compare [9] for the corresponding real hyperbolic result).
In [11], Parker showed that for non-vertical Heisenberg translations there
is aversion of Shimizu’s lemma where the radius of an isometric sphere isbounded in terms of the translation length of the Heisenberg translation at
its centre (see [13] for an analogous result in real hyperbolic space).
This paper summarises the results of [5] which gives ageneralisation ofascrew-parabolic element.
Shimizu’s lemma to groups of isometries ofThis will give analogues of Shimizu’s lemma for all parabolic isometries ofcomplex hyperbolic 2-space. However, our result does not lead to aniceexpression for aprecisely invariant sub-horospherical region.
$\mathrm{H}_{\mathbb{C}}^{n}$$\mathrm{H}_{\mathbb{C}}^{2}$

Screw motions are the most complicated of all parabolic maps. They
combine features of boundary elliptic maps as well as pure parabolic maps.
Indeed, by letting the holonomy tend to the identity ascrew motion tends
to avertical translation and, similarly, letting the translation length tend tozero ascrew motion tends to aboundary elliptic map. Below we will showthat our main theorem interpolates between similar results for these types ofisometry. Specifically, we show that as the holonomy tends to the identity,
our result will tend to Kamiya’s result [6]. On the other hand we will showthat as the translation length goes to zero our result will tend toward theversion of Jorgensen’s inequality for boundary elliptic maps proved in [4].
Our main result depends on anormalisation our screw parabolic map asaparticular Heisenberg screw motion. In the last section of the paper we147
will restate the result to give abound on the radii of isometric spheres interms of the distance of their centres from the axis of the screw-parabolic.
As indicated above, in Proposition 6.4 of [10] it is shown that there is no
precisely invariant horoball for groups containing ascrew-parabolic map. In
this construction, the isometric spheres of very large radius are along way
away from the axis of the screw-parabolic (in particular, it has avery largetranslation length at the centre). Our result will indicate that this conditionis necessary. Specifically, we show that if $A$ is ascrew parabolic map and $B$
is an element of PU $(2, 1)$ with isometric sphere of very large radius $rB$ andif
is discrete, then the centre of the isometric sphere of either $B$ ormust be very far from the axis of $A$ .
This work was begun while the first author was aresearch assistant at
the University of Durham and conclude while the second author was visiting Hunan University. We would like to thank both universities for their
hospitality.
$\langle A, B\rangle$$B^{-1}$2Complex hyperbolic space
We give the necessary background material in this section. More extensive
details can be found in [3].
Letdenote the complex vector space of dimension 3, equipped withanon-degenerate Hermitian form of signature $(2, 1)$ . There are several suchforms. We will use the following form, called the second Hermitian form in$\mathbb{C}^{2,1}$[2]$\langle \mathrm{z}, \mathrm{w}\rangle=z_{1}\overline{w}_{3}+z_{2}\overline{w}_{2}+z_{3}\overline{w}_{1}$,in $u^{1}$, are the column vectors with entries
respectively.
Consider the following subspace in:where$\mathrm{z}$and$\mathrm{w}$$(z_{1}, z_{2}, z_{3})$and$(w_{1}, w_{2}, w_{3})$$\theta^{1},$$V_{0}$$=$$\{\mathrm{z}\in \mathbb{C}^{2,1}-(0,0,0) :$V_{-}$$=$$\{\mathrm{z}\in \mathbb{C}^{2,1} :\langle \mathrm{z}, \mathrm{z}\rangle=0\}$\langle \mathrm{z}, \mathrm{z}\rangle<0\}$.

Let :be the canonical projection onto complex projective space. Thenassociated with the Bergman metric is complex hyperbolic space. The biholomorphic isometry group of
is PU $(2, 1)$
acting by linear projective transformations. Here PU $(2, 1)$ is the projective
unitary group with respect to the Hermitian form defining. This mean$\mathrm{P}$$\mathbb{C}^{2,1}-\{0\}arrow \mathrm{P}\mathbb{C}^{2,1}$$\mathrm{H}_{\mathbb{C}}^{2}=\mathrm{P}(V_{-})$$\mathrm{H}_{\mathbb{C}}^{2}$$\mathbb{C}^{2,1}$$\mathrm{s}$148that the inverse of$B\in \mathrm{P}\mathrm{U}(2,1)$

has the following form

$B=\{\begin{array}{lll}a b cd e fg h j\end{array}\}$,$B^{-1}=$(1)$\overline{\frac{\frac{f}{e}}{d}}\overline{\frac{\frac{c}{b}}{a}}]$The boundary of. It may be identified with the oneispoint compactification of 3-dimensional Heisenberg group , which iswith the group law$\partial \mathrm{H}_{\mathbb{C}}^{2}=\mathrm{P}(V_{0})$$\mathrm{H}_{\mathbb{C}}^{2}$$\mathbb{C}\cross \mathbb{R}$$\mathfrak{R}$$(\zeta_{1},v_{1})0$ $(\zeta_{2},v_{2})=(\zeta_{1}+\zeta_{2},v_{1}+v_{2}+2\Im(\zeta_{1}\overline{\zeta}_{2}))$The identification between$\mathfrak{R}$$\cup\{\infty\}$and$(\zeta, v)-[_{1}^{-|\zeta|^{2}+iv}\sqrt{2}\zeta]\in \mathrm{P}(V_{0})$$\mathrm{P}(V_{0})$,.

is given by$\infty-$$\{\begin{array}{l}100\end{array}\}\in \mathrm{P}(\mathrm{V}0)$

where.
The Heisenberg group has several natural metrics. We choose to workwith the Cygan metric. This is defined by$(\zeta,v)\in \mathfrak{R}$$\rho 0((\zeta_{1},v_{1})$,$(\zeta_{2},v_{2}))=||\zeta_{1}-\zeta_{2}|^{2}+i(v_{1}-v_{2}+2_{S}^{\alpha}(\zeta_{1}\overline{\zeta}_{2}))|^{1/2}$

The Heisenberg group acts on itself by (left) Heisenberg translation: translation
byis given by$T_{(\tau,t)}$$T(\tau,t)$$(\tau,t)\in \mathfrak{R}$: $(z, v)\mapsto(\tau, t)0$ $(z, v)=(z+\tau,v+t+2\Im(\tau\overline{z}))$ .

Heisenberg translation by $(0, t)$ for any given
is called vertical translation by . The unitary group
acts on the Heisenberg group byHeisenberg rotation: rotation
with holonomy
and axis thechain
is given by
$t$

$t$

$\in \mathrm{R}$$\mathrm{U}(1)$$\lambda\in \mathrm{U}(1)$$R_{\lambda,0}$$(0, v)\subset \mathfrak{R}$$R_{\lambda,0}$: $(z, v)\mapsto(\lambda z,v)$ .

, the Heisenberg rotation
For anyby A
with axis
is given by conjugatingby.
The product of aHeisenberg translation and aHeisenberg rotation is aHeisenberg screw motion. The easiest example would be the product ofvertical translation. This iswithgiven by$\tau\in \mathbb{C}$$R_{\lambda,\tau}$$R_{\lambda,0}$$T_{(0,t)}$$S_{\lambda,0,t}$$\in \mathrm{U}(1)$$T_{(\tau,t)}$$R_{\lambda,0}$$S_{\lambda,0,t}$: $(z,v)\mapsto(\lambda z,v+t)$ .

$(\tau, v)\subset \mathfrak{R}$149
It has axis the chain $(0, v)$, rotates about the axis with holonomy
and translates along the axis by aCygan distance,its translation length. Other Heisenberg screw motions may be obtainedby conjugating this one by aHeisenberg translation or by composing otherHeisenberg translations with other Heisenberg rotations.
Heisenberg translations, rotations and screw motions are all isometries of
the Cygan metric, indeed the group of Heisenberg isometries is generated
bywhereand vary overandrespectively.
The action of Heisenberg isometries can be extended to complex hyperbolic space. The Heisenberg translation, Heisenberg rotationand Heisenberg screw motion
correspond to the following matrices in$(2,1)$PU$\subset \mathfrak{R}$$\lambda\in \mathrm{U}(1)$$TR(\tau,t),\lambda,0$$\sqrt{|t|}\in \mathbb{R}$$(\tau, t)$$\lambda$$\mathfrak{R}$$\mathrm{U}(1)$$T_{(\tau,t)}$$R_{(\lambda,0)}$$S_{\lambda,0,t}$$-\sqrt{2}\overline{\tau}01$$T_{(\tau,t)}$$=$$[_{0}^{1}0$$R_{\lambda,0}$$=$$\{\begin{array}{lll}1 0 00 \lambda 00 0 1\end{array}\}$$S_{\lambda,0,t}$$=$$\{\begin{array}{lll}1 0 it0 \lambda 00 0 1\end{array}\}$$-|\tau|^{2}+it\sqrt{2}\tau]1$’,

Elements of PU $(2, 1)$ are classified as parabolic, loxodromic or elliptic just
as for Mobius transformations. An element is called parabolic if and only if
it has aunique fixed point which is in.
$\partial \mathrm{H}_{\mathbb{C}}^{2}$(i) Aparabolic element is called pure parabolic if it is conjugate to a

Heisenberg translation.

(ii) Aparabolic element is called screw parabolic if it is conjugate to a

Heisenberg screw motion.

Suppose that
does not fix , which is equivalent to $g\neq 0$$B$when
has the form (1). Then the isometric sphere (see [10]) of $B$ is thesphere in the Cygan metric with centreand radius $r_{B}=1/\sqrt{|g|}$.
Likewise the isometric sphere of $B$ is the Cygan sphere of radiuswith centre $B(\infty)$ .
$B\in \mathrm{P}\mathrm{U}(2,1)$$\infty$$B^{-1}(\infty)$$1/\sqrt{|g|}$150
3The main results
In this section we summarise the main results of [5].
Theorem 1Suppose that$A$,

have the form

$B\in \mathrm{P}\mathrm{U}(2,1)$$A=\{\begin{array}{lll}1 0 i\mathrm{t}0 \lambda 00 0 \mathrm{l}\end{array}\}$,$B=\{\begin{array}{lll}a b cd e fg h j\end{array}\}$For any real
, A
with $|\lambda-1|<1$ andwhere
let $N(k)$ denote $\max\{|e|^{2}-1, |d|^{2}k, |h|^{2}k, |g|^{2}k^{2}\}$ . If$\mathrm{t}\geq 0$$g\neq \mathrm{E}1$$\in \mathrm{U}(1)$$(N(k)+1)(| \lambda-1|+\frac{t}{k})^{2}<1+\frac{t^{2}}{k^{2}}$then the group$\langle A, B\rangle$

Remark. If we set
to define $N(k)$ to be$k> \frac{2t|\lambda-1|}{1-|\lambda-1|^{2}}$,(2)

is not discrete.

$A=S_{\lambda,\xi,t}$with axis the chain$\max\{|e+\overline{\xi}d\sqrt{2}+\xi h\sqrt{2}+2|\xi|^{2}g|^{2}-1$$(\xi,v)\subset \mathfrak{R}$, $|d+\xi g\sqrt{2}|^{2}k$ ,then we need$|h+\overline{\xi}g\sqrt{2}|^{2}k$,$|g|^{2}k^{2}\}$.

, then is a
We now give two easy corollaries from our theorem. If
vertical translation. In this case, the following corollary shows that Theorem
1reduces to Shimizu’s lemma for vertical Heisenberg translation.
$\lambda=1$$A$

Corollary 2(Theorem 3.2 of [6], Proposition 5.2 of [10]) Suppose
and $B$ in PU $(2, 1)$ are as in Theorem 1with A $=1$ . If

$A$

$0<|g|t<1$ ,then the group$\langle A, B\rangle$

Proof: Choose
we have$k$

is not discrete.

sufficiently large so that $N(k)=|g|^{2}k^{2}$ . Since $|g|t<1$ ,$(N(k)+1) \frac{t^{2}}{k^{2}}<1+\frac{t^{2}}{k^{2}}$.

This is the inequality (2) with $|\lambda-1|=0$ . So Theorem 1implies that thegroup , $B)$ is not discrete.
$\langle$$A$$\square$is aboundary elliptic element. In this case, under theIf $t=0$ , thencondition $|e|>1$ , our theorem reduces to the generalisation of Jorgensen’sinequality for boundary elliptic element of PU $(2, 1)$ given by the authors
$A$

151
Corollary 3(Theorem 5.2(3) of [4]) Suppose A and B in PU(2, 1) are
as in Theorem 1with t $=0$ and $|e|>1$ . If
$0<|e||\lambda-1|<1$ ,then the group$\langle A, B\rangle$

is not discrete.

Since $t=0$ and $|e|>1$ , we can choose asufficiently smallthat $N(k)=|e|^{2}-1$ . Now from $|e||\lambda-1|<1$ , we have$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{f}$$(N(k)+1)|\lambda-1|^{2}<1$$k$so.

This is the inequality (2) with $t=0$ . So Theorem 1iniplies that the groupis not discrete.
$\square$$\langle A, B\rangle$

Finally we restate our main theorem as in an invariant way. Namely, wegive abound on the radii of the isometric spheres of $B$ andin terms of
the Cygan distance between their centres and the axis of .
Given real numbers $x\in[0,1)$ , and $k>2x/(1-x^{2})$ consider the following
functiondefined by$B^{-1}$$A$$\Phi_{x}(k)$$\Phi_{x}(k)=\frac{k(xk+1)}{\sqrt{k^{2}+1}-(xk+1)}$.

The condition $k>2x/(1-x^{2})$ ensures the denominator is positive and so
is positive. For afixed value of , the function
is convex on theinterval $k\in(2x/(1-x^{2}), \infty)$ . Let $k_{0}(x)$ be the value of in this interval
whereattains its minimum value. Using elementary calculus, we seethat $k_{0}(x)$ is aroot of the polynomial$\Phi_{x}(k)$$x$$\Phi_{x}(k)$$k$$\Phi_{x}(k)$$(x^{4}-x^{2})k^{4}+4x^{3}k^{3}+(2x^{2}+x^{4})k^{2}+2(x+2x^{3})k+(1+2x^{2})$.

Theorem 4Let be a screw parabolic element of PU $(2, 1)$ with fixed point, axis
, holonomy
and translation lengthfor somechoice of Cygan metric. Let $B$ be any element of PU $(2, 1)$on
not projectively fixing
and letdenote the radius (with respect to )of the isometric sphere of B. Let$A$$qA\in\partial \mathrm{H}_{\mathbb{C}}^{2}$$L_{A}$$\lambda\in \mathrm{U}(1)$$\sqrt{t}$$\overline{\mathrm{H}_{\mathbb{C}}^{2}}-\{qA\}$$\rho_{0}$$r_{B}$$q_{A}$$R= \max\{\rho_{0}$ $(\mathrm{A}, B(\infty))$,$\rho_{0}(L_{A}, B^{-1}(\infty))$$\mathrm{p}\mathrm{o}$, $\sqrt{tk_{0}(|\lambda-1|)/2}\}$ .

Then$r_{B^{2}}\leq t$ $\Phi_{|\lambda-1|}(2R^{2}/t)=\frac{2R^{2}(2|\lambda-1|R^{2}+t)}{\sqrt{4R^{4}+t^{2}}-(2|\lambda-1|R^{2}+t)}$.

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