Journal of the Institute of Mathematics of Jussieu

NONCOMMUTATIVE REAL ALGEBRAIC GEOMETRY OF KAZHDAN’S PROPERTY (T)

Narutaka Ozawa

RIMS, Kyoto University, 606-8502, Japan (narutaka@kurims.kyoto-u.ac.jp)

Abstract

It is well known that a finitely generated group has Kazhdan’s property (T) if and only if the Laplacian element in has a spectral gap. In this paper, we prove that this phenomenon is witnessed in . Namely, has property (T) if and only if there exist a constant and a finite sequence in such that . This result suggests the possibility of finding new examples of property (T) groups by solving equations in , possibly with the assistance of computers.

(Received January 08 2014)

(Revised July 18 2014)

(Accepted July 18 2014)

2010 Mathematics subject classification:

  • 16A27;
  • 46L89;
  • 22D10;
  • 20F10

Keywords

  • Kazhdan’s property (T);
  • Positivstellensätz;
  • spectral gap

Footnotes

  The author was partially supported by JSPS (23540233).