# Note on non-discrete complex hyperbolic triangle groups of type (n, n, [infinity]; k) II.

1. Introduction. Let n and k be integers greater than 2. Let [I.sub.1],[I.sub.2],[I.sub.3] be the following matrices:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assume that [rho], [sigma], [tau] satisfy the conditions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (the identity matrix) and [I.sub.2][I.sub.3] is a unipotent element. We call the group generated by [I.sub.1], [I.sub.2] and [I.sub.3] a complex hyperbolic triangle group of type (n,n, [infinity]; k) and denote it by [GAMMA](n,n, [infinity]; k). Up to conjugation, there is a one-parameter family of these groups parametrized by k.

It is interesting to ask which values of the parameter correspond to discrete groups as mentioned in .

The purpose of this paper is to show the following theorem, which improves our previous result in  and gives a new list of non-discrete groups of type (n, n, [infinity]; k).

Theorem 1. Let [GAMMA] = ([I.sub.1],[I.sub.2],[I.sub.3]) be a complex hyperbolic triangle group of type (n, n, [infinity]; k) with k [greater than or equal to] [n/2] + 1. The following groups are non-discrete:

(1) [GAMMA](5,5, [infinity]; 3).

(2) [GAMMA](6,6, [infinity]; 5).

(3) [GAMMA](7,7, [infinity]; 4), [GAMMA] (7,7, [infinity]; 5), [GAMMA](7,7, [infinity]; 6).

(4) [GAMMA](8, 8, [infinity]; 5), [GAMMA](8, 8,1;7).

(5) [GAMMA](9, 9, [infinity]; k) for 5 [less than or equal to] k [less than or equal to] 8.

(6) [GAMMA](10,10, [infinity]; k) for 6 [less than or equal to] k [less than or equal to] 9.

(7) [GAMMA](11,11, [infinity]; k) for 6 [less than or equal to] k [less than or equal to] 11.

(8) [GAMMA](12,12, [infinity]; k) for 7 [less than or equal to] k [less than or equal to] 16.

(9) [GAMMA](13,13, [infinity]; k) for 7 [less than or equal to] k [less than or equal to] 38.

(10) [GAMMA](14,14, [infinity]; k) for k [greater than or equal to] 8.

(11) [GAMMA](n, n, [infinity]; k) for any n (> 15).

In  Schwartz classified complex hyperbolic triangle groups into two types. It is said that [GAMMA](n, n, 1) has type B if there is a positive number [k.sub.0] such that [I.sub.1][I.sub.2][I.sub.3] becomes regular elliptic for k > [k.sub.0]. If n [greater than or equal to] 14, then [GAMMA](n, n, [infinity]) has type B. Thus we have:

Corollary 2. If [GAMMA](n,n, [infinity]) has type B, then [GAMMA](n, n, [infinity]; k) is not discrete.

Details for background material on complex hyperbolic space will be found in . For material on complex hyperbolic triangle groups see , , , ,  and .

2. Proof of Theorem 1. To show a group of type (n,n, [infinity]; k) to be non-discrete, we find regular elliptic elements of infinite order.

Lemma 1. Let g be an element of [GAMMA](n, n, [infinity]; k). If trace(g) is real and contained in (-1,3), then g is regular elliptic and trace(g) = 1 + 2 cos [phi][pi]. Moreover, g has finite order if and only if [phi] is a rational number.

In our previous papers [5,8] we used the result by Conway and Jones in . Parker extended their results as follows (see  and ):

Lemma 2 ([10; Theorem A.1.1]). Suppose that we have at most six distinct rational multiples of [pi] lying strictly between 0 and [pi]/2, for which some rational linear combination of their cosines is zero but no proper subset has this property, then the appropriate linear combination is propositional to one of the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We consider the element [I.sub.3][I.sub.1][I.sub.3][I.sub.1][I.sub.2][I.sub.1] in [GAMMA](n, n, [infinity]; k), which is denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is seen that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We note that trace [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is real. It can be shown that the element [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is unipotent in [GAMMA](3,3, [infinity]; k) for any k [greater than or equal to] 4. Also we see that for n [greater than or equal to] 4 the element [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is regular elliptic in [GAMMA](n, n, [infinity]; k) with k [less than or equal to] n - 1 and it is unipotent in [GAMMA](n, n, [infinity]; n).

We are ready to prove our theorem.

Proof of Theorem 1. First we consider the group [GAMMA](21, 21, [infinity]; 18). From the above, the element [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is regular elliptic in [GAMMA](21, 21, [infinity]; 18). Lemma 1 implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has finite order if there is a rational number [phi] satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 2 lists all possible trigonometric Diophantine equations with up to six. We use this result to conclude that there are no rational numbers [phi] satisfying the equation above. It follows from Lemma 1 that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has infinite order in the group [GAMMA](21,21,[infinity];18), which implies that [GAMMA](21, 21,[infinity]; 18) is not discrete.

In the same manner as above, we show that there are no rational numbers [phi] satisfying the following each equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

These correspond to the equations trace [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the following groups, respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It follows that in each group above, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a regular elliptic element with infinite order. Thus the groups above are not discrete. Together with Theorem 2.3 in , we obtain our theorem.

Remark 3. (1) In  Parker, Wang and Xie showed that [GAMMA](3, 3,[infinity]; k) is discrete for k [greater than or equal to] 4. (2) In [GAMMA](13,13,[infinity];8), [GAMMA](11,11,[infinity];7), [GAMMA](7, 7,[infinity];5) among the groups above, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is also regular elliptic.

(3) In [GAMMA](8, 8,[infinity];6), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a regular elliptic element of order 6.

References

 J. H. Conway and A. J. Jones, Trigonometric Diophantine equations (On vanishing sums of roots of unity), Acta Arith. 30 (1976), no. 3, 229-240.

 W. M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999.

 W. M. Goldman and J. R. Parker, Complex hyperbolic ideal triangle groups, J. Reine Angew. Math. 425 (1992), 71-86.

 S. Kamiya, Remarks on complex hyperbolic triangle groups, in Complex analysis and its applications (Osaka, 2007), 219 223, OCAMI Stud., 2, Osaka Munic. Univ. Press, Osaka, 2007.

 S. Kamiya, Note on non-discrete complex hyperbolic triangle groups of type (n,n, [infinity]; k), Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 8, 100-102.

 S. Kamiya, Complex hyperbolic triangle groups of type (n, n, 1), Math. Newsl. 24 (2014), no. 4, 97-103.

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 J. R. Parker, J. Wang and B. Xie, Complex hyperbolic (3,3, n) triangle groups, Pacific J. Math. 280 (2016), no. 2, 433-453.

 R. E. Schwartz, Complex hyperbolic triangle groups, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 339-349, Higher Ed. Press, Beijing, 2002.

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By Shigeyasu KAMIYA

Osaka City University Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

(Communicated by Kenji FUKAYA, M.J.A., June 13, 2017)

2010 Mathematics Subject Classification. Primary 22E40, 32Q45, 51M10.

doi: 10.3792/pjaa.93.67
Author: Printer friendly Cite/link Email Feedback Kamiya, Shigeyasu Japan Academy Proceedings Series A: Mathematical Sciences Report Jul 1, 2017 1406 Twisted Alexander invariants and hyperbolic volume. Award of prizes. Cytokinins