has property (T)

Area of Science:

• Group Theory
• Algebraic Structures
• Computational Mathematics

Background:

• Kazhdan's property (T) is a crucial concept in the study of group representations and locally compact groups.
• Automorphism groups of free groups are fundamental objects in geometric group theory.
• Previous research has explored property (T) for various groups, but its presence in automorphism groups of free groups remained an open question for larger numbers of generators.

Purpose of the Study:

• To determine whether the automorphism group of the free group on 5 generators, Aut(F5), possesses Kazhdan's property (T).
• To provide a constructive and verifiable proof for the presence of property (T) in Aut(F5).

Main Methods:

• Employing a constructive proof methodology.
• Utilizing computer-assisted verification to handle the complexity of the calculations.
• Focusing on the specific structure of Aut(F5) and its relationship to property (T).

Main Results:

• A definitive proof has been established showing that Aut(F5) has Kazhdan's property (T).
• The proof is constructive, meaning it provides a method for demonstrating the property.

Conclusions:

• The automorphism group of the free group on 5 generators, Aut(F5), satisfies Kazhdan's property (T).
• This result advances the understanding of property (T) in the context of automorphism groups of free groups.
• The use of computer-assisted proofs opens avenues for tackling similar complex problems in group theory.