TY - JOUR

T1 - Gehring–Martin–Tan numbers and Tan numbers of elementary subgroups of PSL(2,ℂ)

AU - Maslei, A. V.

PY - 2017/7/1

Y1 - 2017/7/1

N2 - The Gehring–Martin–Tan number and the Tan number are real quantities defined for two-generated subgroups of the group PSL(2,ℂ). It follows from the necessary discreteness conditions proved by Gehring and Martin and, independently, by Tan that, for discrete groups, these quantities are bounded below by 1. In the paper, we find precise values of these numbers for the majority of elementary discrete groups and prove that, for every real r ≥ 1, there are infinitely many elementary discrete groups with the Gehring–Martin–Tan number equal to r and the Tan number equal to r.

AB - The Gehring–Martin–Tan number and the Tan number are real quantities defined for two-generated subgroups of the group PSL(2,ℂ). It follows from the necessary discreteness conditions proved by Gehring and Martin and, independently, by Tan that, for discrete groups, these quantities are bounded below by 1. In the paper, we find precise values of these numbers for the majority of elementary discrete groups and prove that, for every real r ≥ 1, there are infinitely many elementary discrete groups with the Gehring–Martin–Tan number equal to r and the Tan number equal to r.

KW - MOBIUS TRANSFORMATIONS

KW - DISCRETE-GROUPS

KW - KLEINIAN-GROUPS

UR - http://www.scopus.com/inward/record.url?scp=85032349887&partnerID=8YFLogxK

U2 - 10.1134/S0001434617070240

DO - 10.1134/S0001434617070240

M3 - Article

AN - SCOPUS:85032349887

VL - 102

SP - 219

EP - 231

JO - Mathematical Notes

JF - Mathematical Notes

SN - 0001-4346

IS - 1-2

ER -