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Groups graded by root systems and property (T).

Mikhail Ershov1, Andrei Jaikin-Zapirain2, Martin Kassabov3

  • 1Department of Mathematics, University of Virginia, Charlottesville, VA 22904; ershov@virginia.edu.

Proceedings of the National Academy of Sciences of the United States of America
|November 27, 2014
PubMed
Summary
This summary is machine-generated.

Researchers established property (T) for extensive groups, including elementary Chevalley and Steinberg groups, over specific rings. A novel group was constructed that surjects onto finite simple groups of Lie type.

Keywords:
Chevalley groupsSteinberg groupsproperty (T)root system gradings

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Area of Science:

  • Group Theory
  • Algebraic Groups
  • Representation Theory

Background:

  • Property (T) is a significant concept in group theory, related to Kazhdan's property (T).
  • Understanding property (T) in various group classes is crucial for algebraic and geometric applications.

Purpose of the Study:

  • To establish property (T) for a broad category of groups graded by root systems.
  • To construct a new group exhibiting property (T) with specific surjective properties.

Main Methods:

  • Investigating groups graded by root systems, including elementary Chevalley groups and Steinberg groups.
  • Utilizing algebraic and structural properties of these groups over finitely generated commutative rings.

Main Results:

  • Property (T) is established for elementary Chevalley groups and Steinberg groups of rank ≥ 2 over finitely generated commutative rings.
  • A new group with property (T) has been constructed, capable of surjecting onto all finite simple groups of Lie type and rank ≥ 2.

Conclusions:

  • The findings expand the known classes of groups possessing property (T).
  • The constructed group offers a new tool for studying finite simple groups of Lie type.