Chebyshev polynomials and inequalities for Kleinian groups

Hala Alaqad, Jianhua Gong, Gaven Martin

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


The principal character of a representation of the free group of rank two into PSL(2, C) is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of discrete groups and low dimensional topology to determine when such a triple represents a discrete group which is not virtually abelian, that is, a Kleinian group. A classical necessary condition is Jørgensen's inequality. Here, we use certain shifted Chebyshev polynomials and trace identities to determine new families of such inequalities, some of which are best possible. The use of these polynomials also shows how we can identify the principal character of some important subgroups from that of the group itself.

Original languageEnglish
Article number2150102
JournalCommunications in Contemporary Mathematics
Issue number2
Publication statusPublished - Mar 1 2023


  • Chebyshev polynomials
  • Jørgensen's inequality
  • Kleinian groups
  • Principal character

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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