Fuchsian Subgroups of the Picard Group

1976 ◽  
Vol 28 (3) ◽  
pp. 481-485 ◽  
Author(s):  
Benjamin Fine
Keyword(s):  
Free Product ◽  
Semidirect Product ◽  
Finite Index ◽  
Function Theory ◽  
Picard Group ◽  
Automorphic Function ◽  
Linear Transformations ◽  
Gaussian Integers ◽  
Abstract Group ◽  
Generalized Free Product

The Picard group Γ = PSL2 (Z(i)) is the group of linear transformationswith a, b, c, d Gaussian integers.Γ is of interest both as an abstract group and in automorphic function theory [10]. In [10] Waldinger constructed a subgroup H of finite index which is a generalized free product, while in [1] Fine showed that T is a semidirect product with the subgroup H, contained as a subgroup of finite index in the normal factor.

Fuchsian Embeddings in the Bianchi Groups

1987 ◽  
Vol 39 (6) ◽  
pp. 1434-1445 ◽  
Author(s):  
Benjamin Fine
Keyword(s):  
Half Plane ◽  
Modular Group ◽  
Function Theory ◽  
Automorphic Function ◽  
Congruence Subgroups ◽  
Ring Of Integers ◽  
Linear Fractional Transformations ◽  
Upper Half Plane ◽  
Total Structure ◽  
Image Position

If d is a positive square free integer we let Od be the ring of integers in and we let Γd = PSL2(Od), the group of linear fractional transformationsand entries from Od {if d = 1, ad – bc = ±1}. The Γd are called collectively the Bianchi groups and have been studied extensively both as abstract groups and in automorphic function theory {see references}. Of particular interest has been Γ1 – the Picard group. Group theoretically Γ1, is very similar to the classical modular group M = PSL2(Z) both in its total structure [4, 6], and in the structure of its congruence subgroups [8]. Where Γ1 and M differ greatly is in their action on the complex place C. M is Fuchsian and therefore acts discontinuously in the upper half-plane and every subgroup has the same property.

Brownian motion with quadratic killing and some implications

1986 ◽  
Vol 23 (04) ◽  
pp. 893-903 ◽  
Author(s):  
Michael L. Wenocur
Keyword(s):  
Brownian Motion ◽  
Weibull Distribution ◽  
Special Function ◽  
Function Theory ◽  
Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

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