Li, Changjun

Oichi, Makito

Sato, Hiroki

Kyoto University Research Information Repository

Title Jorgensen groups of parabolic type (II) : Countable infinitecase (Perspectives of Hyperbolic Spaces)Author(s)Li, Changjun; Oichi, Makito; Sato, HirokiCitation数理解析研究所講究録 (2003), 1329: 48-57Issue Date2003-06URL http://hdl.handle.net/2433/43253RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityJorgensen groups of parabolic type II(Countable infinite case)(Changjun Li) Shizuoka University(Makito Oichi) Shizuoka University(Hiroki Sato) Shizuoka University ’ABSTRACT. This paper is the second part of the study on Jprgensen groupsof parabolic type. We will show that there are countable infinite many Jprgensengroups of parabolic type on acertain cylinder in this case.1. Introduction.1.1. It is one of the most important problems in the theory of Kleinian groups todecide whether or not asubgroup $G$ of the M\"obius transformation group is discrete.For this problem there are two important and useful theorems: One is Poincare”spolyhedron theorem, which is asufficient condition for $G$ to be discrete. The otheris $\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}$ ’s inequality, which is anecessary condition for atw0-generator M\"obiustransformation group $\langle A, B\rangle$ to be discrete. Here we will consider extreme discretegroups (Jprgensen groups) for Jprgensen’s inequality. This paper is the second part* Partly supported by the Grants-in-Aid for Co operative Hesearch as well as Scientific Research,the Ministry of Education, Science, Sports and Culture, Japan.2000 Mathematics Subject Classification. Primary $30\mathrm{F}40j$ Secondary $20\mathrm{H}10,32\mathrm{G}15$ .Key Words and Phrases. Jprgensen’s inequality, Jprgensen number, Jprgensen group, Kleiniangroup, Poincar\’e’s polyhedron theorem.1329 2003 48-5748of aseries of studies on $\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}$ groups (cf. Li -Oichi -Sato [4]).1.2. Let M\"ob denote the set of all linear fractional transformations (M\"obiustransformations)$A(z)= \frac{az+b}{cz+d}$of the extended complex plane $\hat{\mathrm{C}}=\mathrm{C}\cup\{\infty\}$ , where $a,$ $b,$ $c,$ $d$ are complex numbersand the determinant $ad-bc=1$ . There is an isomorphism between M\"ob and$PSL(2, \mathrm{C})$ . We always write elements of M\"ob as matrices with determinant 1inthis paper. We recall that M\"ob $(=PSL(2, \mathrm{C}))$ acts on the upper half space $H^{3}$ of$\mathrm{R}^{3}$ as the group of conformal isometries of hyperbolic 3-space.In this paper we use aKleinian group in the same meaning as adiscrete group.Namely, aKleinian group is adiscrete subgroup of M\"ob. AKleinian group $G$ is ofthe first kind if the limit set $\Lambda(G)$ of $G$ is all of the extended complex plane$\hat{\mathrm{C}}$ andit is of the second kind otherwise. Asubgroup $G$ of M\"ob is said to be elernentary ifthere exists afinite $G$-orbit in $\mathrm{R}^{3}$ .1.3. The trace $\mathrm{t}\mathrm{r}(A)$ of the matrix$A=(\begin{array}{ll}a bc d\end{array})$ $(ad-bc=1)$in $SL(2, \mathrm{C})$ is defined by $\mathrm{t}\mathrm{r}(A)=a+d$ . We remark that the $\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$ of an elementof M\"ob $(=PSL(2, \mathrm{C}))$ is not well-defined, but Jprgensen number (defined later) isstill well-defined after choosing matrix representatives.1.4. Let $A^{*}$ and $B^{*}$ be matrices in $SL(2, \mathrm{C})$ representing the M\"obius transfor-mations $A$ and $B$ , respectively. As $A^{*}$ and $B^{*}$ are determined by $A$ and $B$ to withinafactor $\mathrm{o}\mathrm{f}-1$ , we see that the commutator $A^{*}B^{*}(A^{*})^{-1}(B^{*})^{-1}$ (resp. $(A^{*})^{2}$ ) areuniquely determined by $A$ and $B$ (resp. $A$). Thus we may write $\mathrm{t}\mathrm{r}(ABA^{-1}B^{-1})=$$\mathrm{t}\mathrm{r}(A^{*}B^{*}(A^{*})^{-1}(B^{*})^{-1})$ and $\mathrm{t}\mathrm{r}^{2}(A)=\mathrm{t}\mathrm{r}^{2}(A^{*})$ .49In 1976 Jprgensen obtained the following important theorem, which gives anec-essary condition for anon-elementary M\"obius transformation group $G=\langle A, B\rangle$ tobe discrete.Theorem A(Jprgensen [1]). Suppose that the M\"obius transformations $A$ and$B$ generate a $nonarrow elementary$ discrete group. Then$J(A, B):=|\mathrm{t}\mathrm{r}^{2}(A)-4|+|\mathrm{t}\mathrm{r}(ABA^{-1}B^{-1})-2|\geq 1$ .The lower bound 1is best possible.1.5. DEFINITION 1. Let $A$ and $B$ be M\"obius transformations. The Jlrgensen$num,berJ(A, B)$ for the ordered pair $(A, B)$ is defined by$J(A, B):=|\mathrm{t}\mathrm{r}^{2}(A)-4|+|\mathrm{t}\mathrm{r}(ABA^{-1}B^{-1})-2|$ .DEFINITION 2. Asubgroup $G$ of M\"ob is called aJlrgensen group if $G$ satisfiesthe following four conditions:(1) $G$ is atw0-generator group.(2) $G$ is adiscrete group.(3) $G$ is anon-elementary group.(4) There exist generators $A$ and $B$ of $G$ such that $J(A, B)=1$ .1.6 J$rgensen and Kiikka showed the following.Theorem $\mathrm{B}$ (Jergensen-Kiikka [2]). Let $\langle A, B\rangle$ be a non-elernentary discretegrvup with $J(A, B)=1$ . Then $A$ is elliptic of orvler at least seven or $A$ is parabolic.If $\langle A, B\rangle$ is aJdrgensen group such that $A$ is parabolic and $J(A, B)=1$ , thenwe call it a Jprgensen group of parabolic type. There are infinite many JPrgensengroups of parabolic type (J$rgensen-Lascurain-Pignataro [3], Sato [6]).Now it gives rise to the following problem.50Problem 1. Find all Jmrgensen groups of parabolic type.1.7. Let $\langle A, B\rangle$ be amarked tw0-generator group such that $A$ is parabolic.Then we can normalize $A$ and $B$ as follows:$A=(\begin{array}{ll}1 10 1\end{array})$ and $B_{\sigma,\mu}=(\begin{array}{lll}\mu\sigma \mu^{2}\sigma -1/\sigma\sigma \mu\sigma\end{array})$where $\sigma\in \mathrm{C}\backslash \{0\}$ and $\mu\in \mathrm{C}$ . See [4] for this normalization.We denote by $G_{\sigma,\mu}$ the marked group generated by $A$ and $B_{\sigma,\mu}:G_{\sigma,\mu}=(\mathrm{A}$ $B_{\sigma,\mu}\rangle$ .We say that $(\sigma,\mu)\in \mathrm{C}\backslash \{0\}\cross \mathrm{C}$ is the point representing a marked group $G_{\sigma,\mu}$ andthat $G_{\sigma,\mu}$ is the marked group corresponding to a point $(\sigma, \mu)$ .1.8. In [6], Sato considered the case of $\mu=r,\cdot k(k\in \mathrm{R})$ . Namely, he consideredmarked tw0-generator group $G_{\sigma,\mathrm{s}k}=\langle A, B_{\sigma,ik}\rangle$ generated by$A=(\begin{array}{ll}1 10 1\end{array})$ and $B_{\sigma_{1}ik}=(\begin{array}{lll}ik\sigma -k^{2}\sigma -1/\sigma\sigma ik^{\wedge}\sigma \end{array})$where $\sigma\in \mathrm{C}\backslash \{0\}$ and $k\in \mathrm{R}$ .Now we have the following conjecture.CONJECTURE. For any Jergensen group $G$ of parabolic type there exists amarked group $G_{\sigma,.k}.(\sigma\in \mathrm{C}\backslash \{0\}, k\in \mathrm{R})$ such that $G_{\sigma,ik}$ is conjugate to $G$ .If this conjecture is true, then it is sufficient to consider the case of $\mu=ik$ inorder to find all Jirgensen groups of parabolic type. In this paper we only considerthe case of $\mu=ik$ .1.9. Let $C$ be the following cylinder:$C=$ { $(\sigma,$ $ik)||\sigma|=1,$ &E $\mathrm{R}$ }.Theorem $\mathrm{C}$ (Sato [6]). If a marked twO-generator grvup $G_{\sigma,ik}(\sigma\in \mathrm{C}\backslash \{0\},$ $k\in$51R) is a Jprgensen group, then the point $(\sigma,$ik) representing $G_{\sigma,ik}$ lies on the cylinderC.By Theorem $\mathrm{C}$ , if $(\theta, k)$ is apoint on the cylinder $C$ , then we can set $\sigma=$$-ie^{j\theta}(0\leq\theta\leq 2\pi)$ . If apoint $(-ie^{j\theta}, ik)$ on the cylinder $C$ represents a $\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}$group, then we say that the group is aJprgensen group of parabolic type $(\theta, k)$ .Now it gives rise to the following problem.Problem 2. Find all Jergensen groups of parabolic type $(\theta, k)$ .We divide Jprgensen groups of this type into three parts as follows:Part 1. $|k|\leq\sqrt{3}/2,0\leq\theta\leq 2\pi$ (finite case).Part 2. $\sqrt{3}/2<|k|\leq 1,0\leq\theta\leq 2\pi$ (countable infinite case).Part 3. $1<|k|,$ $0\leq\theta\leq 2\pi$ (uncountable infinite case).By some lemmas in [6], it suffices to consider the case of $0\leq\theta\leq\pi/2$ and $k\geq 0$in order to find Jmrgensen groups of parabolic type $(\theta,k)$ .In the previous paper [4] we find all Jprgensen groups in the case where $0\leq\theta\leq$$\pi/2$ and $0\leq k\leq\sqrt{3}/2$ , that is, we obtain the following theorem.Theorem $\mathrm{D}$ (finite case) (Li -Oichi -Sato [4]). (i) There are sixteenJprgensen grvyups in $D=\{(\theta, k)\in \mathrm{R}|0\leq\theta\leq\pi/2,0\leq k\leq\sqrt{3}/2\}$.(ii) Nine of them are Kleinian groups of the first kind and seven groups are of thesecond kind.1.10. For asufficient condition for asubgroup of the M\"obius transformationgroup to be discrete, the following theorem is well-known.Theorem $\mathrm{E}$ (Poincare”sPolyhedron Theorem (Maskit [5, p.73])).Let $P$ be a polyhedron with side pairing transfomations satisfying the followingconditions (1) $th,rough(6)$ . Then, $G$ , the group generated by the side pairing trans-52forrnations, is discrete and $P$ is a fundarnental polyhedron for $G_{f}$ and the refiectionrelations and cycle relations forrn a complete set of relations for $G$ :(1) For each side $s$ of $P$ , there is a side $s’$ and there is an element $g_{s}\in G$satisfying $g_{s}(s)=s’$ and $g_{s’}=g_{s}^{-1}$ .(2) $g_{s}(P)\cap P=\emptyset$ .(3) For every point $z\in P^{*},p^{-1}(z)$ is a finite set. Here $P^{*}$ is the space ofequivalence classes so that the projection $p$ : $\overline{P}$ (the closure of $P$) $arrow P^{*}$ is continousand open.(4) Let $e$ be an edge and let $h$ be the cycle transformation at $e$ . Then for eachedge $e_{J}$ there is a positive integer $t$ such that $h^{t}=1$ .(5) Let $\{e_{1}, e_{2}, \ldots, e_{m}\}$ be any cycle of edges of $P$ and let $\alpha(e_{k})(k=1,2, \ldots, m)$be the angle rneasured frorn inside $P$ at the edge $e_{h}$ . Let $q$ be the srnallest positiveinteger such that $h^{q}=1$ , where $h$ is the cycle transforrnation at $e_{1}$ . Th.en the equality$\sum_{k=1}^{m}\alpha(e_{k})=2\pi/q$holds.(6) $P^{*}$ is cornplete.2. Theorems.In this section we will state that we find all Jergensen groups in Part 2, that is,we obtain the following theorems. The proofs will appear elsewhere.Main Theorem. There are countable infinite rnany Jorgensen groups on theregion 1 $(\theta, k)|0\leq\theta\leq\pi/2,$ $\sqrt{3}/2<k\leq 1\}$ .For simplicity we write $B_{\theta,k}$ for $B_{-\cdot e^{j\theta},ik}.$ .This theorem consists of the following Theorem 1through Theorem 6.53Let $A$ and $B_{\theta,k}(k\in \mathrm{R}, 0\leq\theta\leq\pi/2)$ be the following matrices:$A=(\begin{array}{ll}1 \mathrm{l}0 1\end{array})$ and $B_{\theta,k}=(\begin{array}{lll}ke^{i\theta} ie^{-i\theta}(k^{2}\wedge e^{2i\theta} -1)-ie^{i\theta} ke^{i\theta} \end{array})$We can prove these theorems by using Jprgensen’s inequality and Poincare”spolyhedron theorem.Theorem 1(Li -Oichi -Sato [4]). Let $G_{\theta,k}=\langle A, B_{\theta,k}\rangle$ be the group gener-ated by $A$ and $B_{\theta,k}$ . If $0<\theta<\pi/6,$ $\pi/6<\theta<\pi/4,$ $\pi/4<\theta<\pi/3,$ $\pi/3<\theta<\pi/2$ ,then $G_{\theta,k}=\langle A, B_{\theta,k}\rangle$ are not Kleinian groups and so not Jlrgensen groups for $it\in \mathrm{R}$ .Theorem 2(the case of $\theta=0$ ). Let$A=(\begin{array}{ll}\mathrm{l} 10 1\end{array})$ and $B_{h}:=B_{0,k}=(\begin{array}{ll}k i(k^{2}-1)-i k\end{array})(k\in \mathrm{R})$ ,and let $G_{k}=\langle A, B_{k}\rangle$ . Then the following hold.(i) In the case where $\cos(\pi/2m)<k<\cos(\pi/(2m+2))$ and $k\neq\cos(\pi/(2m+1))$$(|n=3,4, \cdots)_{f}G_{k}$ are not Kleinian groups and not Jprgensen groups.(ii) In the case of $k=1,$ $G_{k}$ is a Kleinian group of th.e second kind and $a$Jprgensen group, and $\Omega(G_{k})/G_{k}$ is a union of two Riemann surfaces with signature$($0; 2, 3, $\infty)$ .(iii) In the case of $k=\cos(\pi/n)(n=7,8, \ldots),$ $G_{k}$ is a Kleinian group ofthe second kind and a Jlrgensen group, and $\Omega(G_{k})/G_{k}$ is a union of two Riernannsurfaces with signatures $($0; 2, 3, $n)$ and $($ 0; 2, 3, $\infty)$ .Theorem 3(the case of $\theta=\pi/6$ ). LetA $=(\begin{array}{ll}1 \mathrm{l}0 1\end{array})$ and $B_{k}$. $:=B_{\pi/6,k}=(\begin{array}{lll}ke^{\pi i/6} i(k^{2}e^{\pi\dot{\cdot}/6} -e^{-\pi})j/6-ie^{\pi i/6} ke^{\pi i/6} \end{array})(k\in \mathrm{R})$ ,54an.d let $G_{k}=\langle A, B_{k}\rangle$ . Then $G_{h}$ are not Kleinian groups and not Jprgensen groupsfor&with $\sqrt{3}/2<k\leq 1$ .Theorem 4(the case of $\theta=\pi/4$ ). Let$A=(\begin{array}{ll}1 10 1\end{array})$ and $B_{k}:=B_{\pi/4,k}=(\begin{array}{ll}ke^{\pi\dot{\cdot}/4} .-e^{-\pi i/4})i(k^{2}e^{\pi\cdot/4}-ie^{\pi i/4} ke^{\pi\cdot/4}\end{array})(k\in \mathrm{R})$ ,and let $G_{k}=\langle A, B_{k}\rangle$ . Then the following h.old.(i) In case of $\sqrt{3}/2<k<1,$ $G_{h}$ are not Kleinian groups and not Jprgensengroups.(ii) In the case of $k=1,$ $G_{k}$ is a Kleinian group of the first kind and a Jprgensengroup. The volume $V(G_{\pi/4,1})$ of the 3-orbifold for $G_{\pi/4,1}$ is$V(G_{\pi/4,1})=8[2L(\pi/4)-L(\pi/12)-L(5\pi/12)]$ ,where $L(\theta)$ is the Lobachevskii function:$L( \theta)=-\int_{0}^{\theta}\log|2\sin u|du$ .Theorem 5(the case of $\theta=\pi/3$ ). Let$A=(\begin{array}{ll}1 10 1\end{array})$ and $B_{k}:=B_{\pi/3,h}=(\begin{array}{ll}ke^{\pi i/3} -e^{-\pi i/3})i(k^{2}e^{\pi i/3}-ie^{\pi|/3} ke^{\pi\cdot/3}\end{array})(k\in \mathrm{R})$,and let $G_{k}=\langle A, B_{k}\rangle$ . Then $G_{k}$ are rtot Kleinian groups and not Jprgensen groupsfor $k$ with $\sqrt{3}/2<k\leq 1$ .Theorem 6(the case of $\theta=\pi/2$ ). Let$A=(\begin{array}{ll}\mathrm{l} 10 1\end{array})$ aanndd $B_{h}:=B_{\pi/2,\kappa=}.(\begin{array}{lll}ik^{\wedge} -(k^{2} +1)\mathrm{l} ik^{\wedge} \end{array})(k\in \mathrm{R})$,55and let $G_{k}=\langle A, B_{k}\rangle$ . Then the following hold.(i) In the case where $\cos(\pi/(2n-1))<k<\cos(\pi/(2n+1))$ and $k\neq\cos(\pi/2n)$ $($$n=3,4,$ $\ldots),$ $G_{k}$ are not Kleinian groups and not Jprgensen groups.(ii) In the case of $k=1,$ $G_{h}$ is a Kleinian group of the second kind and $a$Jprgensen $group_{f}$ and $\Omega(G_{k})/G_{k}$ is a Riemann surface with signature $($0; 3, 3, $\infty)$ .(iii) In the case of $k=\cos(\pi/n)(n=7,8, \ldots),$ $G_{k}$ are Kleinian groups ofthe second kind and Jprgensen groups, and $\Omega(G_{h})/G_{k}$ is a Riemann surface withsignature $($ 0; 3, 3, $n)$ .References[1] T. J$rgensen, On discrete grvups of M\"obius transforrnations, Amer. J. Math.98 (1976), 739-749.[2] T. Jlrgensen and M. Kiikka, Some extreme discrete grvups, Ann. Acad. Sci.Fenn. 1(1975), 245-248.[3] T. $\mathrm{J}\emptyset \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}$ , ALascurain and T. Pignataro, Translation extensions of theclassical modular group, Complex Variables 19 (1992), 205-209.[4] C. Li, M. Oichi and H. Sato, Jgrgenseri groups of parabolic type I (Finite case),preprint.[5] B. Maskit, Kleinian Groups, Springer-Verlag, New York, Berlin, Heidelberg,1987.[6] H. Sato, One-parameter families of extreme groups for Jprgensen’s inequality,Contemporary Math. 256 (The First Ahlfors -Bers Colloquium) edited by I.Kra and B. Maskit, 2000, 271-287.56[7] H. Sato and R. Yamada, Some extreme Kleinian groups for $Jprgensen^{f}s$ in-equality, Rep. Fac. Sci. Shizuoka Univ. 27 (1993), 1-8.Department of mathematics Faculty of scienceShizuoka University836 Ohya, Shizuoka 422-8529Japan$\mathrm{e}$-mail:r5144012@ipc.shizuoka.ac.jpe-mail:r5544012@ipc.shizuoka.ac.jpe-mail:smhsato@ipc.shizuoka.ac.jp57

Jorgensen groups of parabolic type (II) : Countable infinite case (Perspectives of Hyperbolic Spaces)

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Jorgensen groups of parabolic type (II) : Countable infinite case (Perspectives of Hyperbolic Spaces)

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