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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.
Summary
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.
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.
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