数理解析研究所講究録 1223 巻 2001 年 56-60

56ofNotes on discrete subgroups
PU(1,2;c)with Heisenberg translations III
Shigeyasu KAMIYA
神谷茂保 (岡山理大)

In a previous paper [8] we have seen that under some conditions Parker’s theorem
yields the discreteness condition of Basmajian and Miner for groups with aHeisenberg
translation. In this paper we give a new stable basin region and show the same result as
in [8] without the assumption on r. This is ajoint work with John R. Parker.
1. First we recall some definitions and notation. Let
be the field of complex
numbers. Let $V=V^{1,2}(\mathrm{C})$ denote the vector space , together with the unitary structure
defined by the Hermitian form$\mathrm{C}$$\mathrm{C}^{3}$$\tilde{\Phi}(z^{*},w^{*})=-\cdot(\overline{z_{0}^{*}}w_{1}^{*}+\overline{z_{1}^{*}}w_{0}^{*})+\overline{z_{2}^{*}}w_{2}^{l}$forin . An automorphism of , that is alinear
bijection such thatfor
in , will be called aunitarytransformation. We denote the group of all unitary transformations by $U(1,2;\mathrm{C})$ . Let$V_{0}=$,to’) $<0$ }. It is clear that
{ $w^{*}\in V|$ ( , to’) $=0$ } and $V_{-}=$ { $w^{*}\in V|$$U(1,2;\mathrm{C})$$U(1,2;\mathrm{C})/(center)$andare invariant under. We denoteby PU 1, 2;. Set$V^{*}=V_{-}\cup V_{0}-\{0\}$ .Let
: $V^{*}arrow\pi(V^{*})$ be the projection map defined by, whereand. We write for $\pi(0,1,0)$ . We may identify
with the Siegel domain$z’=(z_{0}^{*}, z_{1}^{*},z_{2}^{*}),w^{*}=(w_{0}^{*},w_{1}^{*},w_{2}^{*})$$V$$\tilde{\Phi}(g(z^{*}),g(w^{*}))=\tilde{\Phi}(z^{*},w^{*})$$\tilde{\Phi}$$w^{*}$$V$$g$$z^{*},w^{*}$$V$$\tilde{\Phi}(w^{*}$$V_{0}$$V_{-}$$($$\pi(w_{0}^{*},w_{1}^{*},w_{2}^{*})=$$\pi$$(w_{1}, w_{2})$$w_{1}=w_{1}^{*}/w_{0}^{*}$$\mathrm{C})$$w_{2}=w_{2}^{*}/w_{0}^{*}$$\infty$$\pi(V_{-})$$H^{2}= \{w=(w_{1},w_{2})\in \mathrm{C}^{2}| Re(w_{1})>\frac{1}{2}|w_{2}|^{2}\}$.

We can regard an element of PU(1, 2; CLas transformation acting on
and its boundary(see [6]). Denoteby. We define anew coordinate system in. Our convention slightly differs from Basmajian-Miner [1] and Parker [8]. The $H-$
coordinates of apoint
are defined by$k=Re(w_{1})\frac{1}{2}|w_{2}|^{2}$such thatand $t=Im(w_{1})$ . For simplicity, we write $(t_{1},w’)_{H}$ for$(0, t_{1},w’)_{H}$ . The Cygan metric $\rho(p,q)$ for $p=(k_{1},t_{1},w’)_{H}$ and $q=(k_{2},t_{2},W’)_{H}$ is given
by$H^{2}$$H^{2}\cup\partial H^{2}$$\partial H^{2}$$\overline{H^{2}}-$$H^{2}$$\{\infty\}$$(w_{1},w_{2})\in\overline{H^{2}}-\{\infty\}$$(k, t,\mathrm{W}2)\mathrm{t}\in(\mathrm{R}^{+}\mathrm{U}\{0\})\cross \mathrm{R}\cross \mathrm{C}$$\rho(p,q)=|\{\frac{1}{2}|W’-w’|^{2}+|k_{2}-k_{1}|\}+i\{t_{1}-t_{2}+Im(\overline{w’}W’)\}|^{*}$

We note that the Cygan metric$’-1\backslash$$\rho$.

is ageneralization of the Heisenberg metric$\delta$in$\partial H^{2}$

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Let f

$\ovalbox{\tt\small REJECT}$(be an element of PU (1,of f byC) with $f(-)+$ – We define the$2\ovalbox{\tt\small REJECT}$$a_{ij^{\ovalbox{\tt\small REJECT}}}S\ovalbox{\tt\small REJECT} i,j\ovalbox{\tt\small REJECT} S$

isometric sphere$r_{f}$,$I_{f}=\{w=(w_{1}, w_{2})\in\overline{H}^{2}| |\tilde{\Phi}(W, Q)|=|\tilde{\Phi}(W, f^{-1}(Q))|\}$(see [4]). It follows that the isometric sphere
where $Q=(0,1,0)$ , $W=(1,w_{1},w_{2})$ inis the sphere in the Cygan metric with centerand radius $Rf=\sqrt{1/|a_{12}|}$, that,$V^{*}$$f^{-1}(\infty)$$i\mathrm{s}If$$I_{f}=\{z=(k,t, w’)_{H}\in(\mathrm{R}^{+}\cup\{0\})\cross \mathrm{R}\cross \mathrm{C}|\rho(z, f^{-1}(\infty))=\sqrt{\frac{1}{|a_{12}|}}\}$.

2. We shall give amodified version of the stable basin theorem in [8]. Let$B_{r}=\{z\in\partial H^{2}|\delta(z,0)<r\}$,. Given and with $<s$ , the pair of open sets
and letis said$g\in
S$ ,betostable with respect to aset of elements in PU(1, 2; C) if for any element$\overline{B}_{s}^{c}=\partial H^{2}-\overline{B}_{s}$$r$$(B_{r},\vec{B_{s}})$

$r$

$s$

$S$

$g(0)\in B_{r}$$g(\infty)\in\overline{B}_{s}^{\mathrm{c}}$.

Aloxodromic element has aunique complex dilation factor
such that $|\lambda(f)|>1$ .
Letdenote the family of loxodromic elements with fixed points in,a$r<1/\sqrt{3}$$|\lambda(f)-1|<\epsilon$
and satisfying. For positive real numbers and withand $r’<1$ ,we defineby$f$$\lambda(f)$$S(r, \epsilon)$$f$$B_{r}$$\mathrm{d}$$\vec{B_{1/\mu}}$

$r’$

$r$

$\epsilon(r, r’)$$\epsilon(r,r’)=\sup\{|\lambda(f)-1|\}$,(2.1)

where $|\lambda(f)-1|$ satisfies the inequality.

$|\lambda(f)-1|<\sqrt{1+(\frac{1-(3+|\lambda(f)-1|)r^{2})}{1-2r^{2}})^{2}(\frac{1-3r^{2}}{1-r^{2}})^{2}(\frac{r’}{r})^{2}}-1$(2.2)

Atriple of non-negative numbers
is said to be abasin point provided that $r<1/\sqrt{3}$,$r’<1$ and $<\epsilon(r, r’)$ . In particular, if $r’\leq r$ , we callastable basin point. Callthe set of all such points the stable basin region. For simplicity, we abbreviate
to. Figure 1shows our new stable basin region, which contains regions in [1] and [8].
Some stable basin points are tabulated in Table 2.
$(r,\mathrm{r}’,\epsilon)$$(r, r’,\epsilon)$$\epsilon$$(r,r,\epsilon)$$(r, \epsilon)$

Exactly the same arguments except for using the following Lemma 2.1 instead of
Proposition 3.3 in [1] shows our new stable basin theorem.

Lemma 2.1. Let , $c>0$ be given. If$b$factor satisfies$f$$|\lambda(f)-1|\leq\sqrt{1+(b/c)^{2}}-1$

is a complex dilation and its complex dilation, then $f(p)\in B_{b}(p)$ for.
$p\in\overline{B}_{\mathrm{c}}$58
Theorem 2.2 (cf. [8; Stable Basin Theorem]). Given positive real numbers r andwith r
and $r’<1$ , the pair of open sets
is stable with respect to the, wherefamily
is given ) (i.i).
$r^{\mathrm{F}}$$<x\mathit{7}$$\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}^{\ovalbox{\tt\small REJECT}}$$(B_{\mathrm{r}^{\ovalbox{\tt\small REJECT}_{\mathrm{t}}}}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT} 7^{\mathrm{r}^{\mathrm{t}}}})$$\ovalbox{\tt\small REJECT} 5(\mathrm{r},\mathrm{g}(\mathrm{r},\mathrm{r}’))$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$\mathrm{g}(\mathrm{r}, \mathrm{r}’)$Remark 2.3. By arguing as in Corolary 6.14 in [1], we may find the boundary of thestable basin region by equating both sides of inequality (2.2) and solving for $|\lambda(f)-1|$ interms of . If we use Basmajian and Miner’s inequality (6.2) in [1], this involves solving apolynomial of degree 6. Using our inequality (2.2), we have$r$$| \lambda(f)-1|<\frac{a_{2}a_{1}\sqrt{a_{3}^{4}b_{1}+a_{2}^{2}a_{1}^{2}+2a_{3}^{3}r^{2}b_{1}}-a_{2}^{2}a_{1}^{2}-a_{3}^{3}b_{1}r^{2}}{a_{2}^{2}a_{1}^{2}-a_{3}^{2}b_{1}r^{4}}$,where $a_{j}=1-jr^{2}$ and for $j=1,2,3$ and $b_{1}=(r’/r)^{2}$ .

3. We begin with recalling Parker’s theorem on the discreteness of subgroups ofPU 1, 2; .
$($$\mathrm{C})$Theorem 3.1 ([9; Theorem 2.1]). Let$g$be a Heisenberg translation with the for$m$$g=(\begin{array}{lll}1 0 0s \mathrm{l}\overline{a} a 0 1\end{array})$

where $Re(s)= \frac{1}{2}|a|^{2}$ . Let$R_{f}$. If$f$,be any element of PU(1, 2; C) with isometric sphere of radius$R_{f}^{2}>\delta(gf^{-1}(\infty), f^{-1}(\infty))\delta(gf(\infty), f(\infty))+2|a|^{2}$

then$\theta\iota e$group $<f,g>generated$ by$f$

and$g$,

is not discrete.

In Theorem 4.5 of [8] we have shown that if $r<0.484$ , then Theorem 3.1 leads to thediscreteness condition of Basmajian and Miner for groups with aHeisenberg translation.
By using amore precise estimate on the Heisenberg distance between fixed points of interms of, we have the following same result without the assumption on .
and$f$$R_{f}$$\lambda(f)$$r$

Theorem 3.2. Fix a stable basin point. Let be the same element as in Theorem
S. 1. Let be a loxodromic element with fixed point 0and , and satisfying.
$If\delta(0,q)>\mathrm{m}_{r}^{\iota 0,0}(1+r^{2}+\sqrt{1+r^{2}})$, then the group $<f,g>generated$ by and is notdiscrete.
$(r,\epsilon)$

$g$

$f$

$|\lambda(f)-1|<\epsilon$

$q$

$f$

$g$

To prove our theorem, we need the following lemmas.

Lemma 3.3. Let. then$f$be a loxodromic element with fixed points 0and , $satisfying|\lambda(f)-$$q$$1|<\epsilon$.

$( \frac{\delta(0,q)}{R_{f}})^{2}\leq\frac{2\epsilon-\epsilon^{2}}{1-\epsilon}$

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Lemma 3.4. For a stable basin point$(r,\epsilon)$,$\frac{1+r^{2}+\sqrt{1+r^{2}}}{r^{2}}>(\frac{2\epsilon-\epsilon^{2}}{1-\epsilon})^{\#}(2+(8+\frac{M(\in)}{2})^{*})$

where $M(\epsilon)=(1+\epsilon)^{*}+(1+\epsilon)^{-*}$ .

Figure 1. Graph ofTable 2$\epsilon(r,r)$,

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References
1. A. Basmajian and R. Miner, Discrete subgroups of complex hyperbolic motions, Invent. Math. 131, 85-136 (1998).
2. A.F. Beardon, The Geometry of Discrete Groups, Springer-Verlag, New York, 1983.
3. L. R. Ford, Automorphic Functions (Second Edition), Chelsea, New York, 1951.
4. W. M. Goldman, Complex hyperbolic geometry, Oxford University Press, 1999.
5. S. Kamiya, Notes on non-discrete subgroups of (1, n;F), Hiroshima Math. J. 13,501-506, (1983).
6. S. Kamiya, Notes on elements of $U(1,n;$ C), Hiroshima Math. J. 21, 23-45, (1991).
7. S. Kamiya, Parabolic elements of $U(1,n;$ C), Rev. Romaine Math. Pures et Appl. 40,55-64, (1995).
8. S. Kamiya, On discrete subgroups of PU(1,2; C) with Heisenberg translations, J.
London Math. Soc. (2) 62, 827-842 (2000).
9. J. Parker, Uniform discreteness and Heisenberg translations, Math. Z. 225, 485-505(1997).
$\tilde{U}$Okayama University of Science
1-1 Ridai-cho, Okayama 700-0005JAPAN-mail:kaniya@mech.ous.ac.jp$\mathrm{e}$