scholarly journals Singular Sets of Some Kleinian Groups

1966 ◽  
Vol 26 ◽  
pp. 127-143 ◽  
Author(s):  
Tohru Akaza
Keyword(s):  
Kleinian Groups ◽  
Schottky Groups ◽  
Dimensional Measure ◽  
Singular Sets

In our paper Cl] we showed that there exist Schottky groups whose singular sets have positive 1-dimensional measure. Since the example was very complicated, it is natural to seek for simpler examples. Further the problem how about the singular sets of more general groups occurs.

scholarly journals Singular Sets of Some Kleinian Groups (II)

1967 ◽  
Vol 29 ◽  
pp. 145-162 ◽  
Author(s):  
Tohru Akaza
Keyword(s):  
Kleinian Groups ◽  
Singular Set ◽  
Fundamental Domains ◽  
Dimensional Measure ◽  
Singular Sets ◽  
Discontinuous Groups ◽  
Positive Capacity ◽  
Automorphic Functions

In the theory of automorphic functions it is important to investigate the properties of the singular sets of the properly discontinuous groups. But we seem to know nothing about the size or structure of the singular sets of Kleinian groups except the results due to Myrberg and Akaza [1], which state that the singular set has positive capacity and there exist Kleinian groups whose singular sets have positive 1-dimensional measure. In our recent paper [2], we proved the existence of Kleinian groups with fundamental domains bounded by five circles whose singular sets have positive 1-dimensional measure and presented the problem whether there exist or not such groups in the case of four circles. The purpose of this paper is to solve this problem. Here we note that, by Schottky’s condition [4], the 1-dimensional measure of the singular set is always zero in the case of three circles.

scholarly journals Poincaré Theta Series and Singular Sets of Schottky Groups

1964 ◽  
Vol 24 ◽  
pp. 43-65 ◽  
Author(s):  
Tohru Akaza
Keyword(s):  
Essential Role ◽  
Theta Series ◽  
Schottky Group ◽  
Singular Set ◽  
Schottky Groups ◽  
Linear Transformations ◽  
Convergence Problem ◽  
Discontinuous Group ◽  
Singular Sets ◽  
Metrical Property

In the theory of automorphic functions for a properly discontinuous group G of linear transformations, the Poincaré theta series plays an essential role, since the convergence problem of the series occupies an important part of the theory. This problem was treated by many mathematicians such as Poincaré, Burnside [2], Fricke [4], Myrberg [6], [7] and others. Poincaré proved that the (-2m)-dimensional Poincaré theta series always converges if m is a positive integer greater than 2, and Burnside treated the problem and conjectured that ( -2)-dimensional Poincaré theta series always converges if G is a Schottky group. This conjecture was solved negatively by Myrberg. As is shown later (Theorem A), the convergence of Poincaré theta series gives an information on a metrical property of the singular set of the group.

scholarly journals Singular sets of some infinitely generated Kleinian groups

1975 ◽  
Vol 26 (4) ◽  
pp. 485-497
Author(s):  
Tohru Akaza ◽  
Eiichi Sakai
Keyword(s):  
Kleinian Groups ◽  
Singular Sets
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