On the profinite rigidity of free and surface groups

Area of Science:

• Group Theory
• Algebraic Topology
• Geometric Group Theory

Background:

• The study of finitely generated groups and their properties is central to modern algebra.
• Residually-p and residually finite groups have significant applications in various mathematical fields.
• Profinite completions offer a powerful tool for understanding group structure.

Purpose of the Study:

• To generalize Baumslag's result on parafree groups to a broader class of residually-p groups.
• To investigate profinite rigidity and confirm Remeslennikov's conjecture for a specific class of groups.
• To establish new criteria for the structure of subgroups within residually-(torsion-free nilpotent) groups.

Main Methods:

• Utilizing the concept of pro-p completions of groups.
• Developing a new ingredient concerning the pro-p topology on virtually polycyclic subgroups.
• Applying techniques from geometric group theory and the theory of profinite groups.

Main Results:

• Demonstrating that two-generated subgroups of finitely generated residually-p groups with the same pro-p completion as free or surface groups are free.
• Proving that for a class of groups with finite abelian hierarchy, a finitely generated residually finite group is isomorphic to its profinite completion.
• Establishing the profinite rigidity of a specific group within the class of finitely generated residually free groups.

Conclusions:

• The findings extend existing theorems in group theory and provide new insights into the structure of subgroups.
• The confirmation of Remeslennikov's conjecture for a specific class of groups advances the understanding of profinite rigidity.
• This research contributes to the broader study of group structure, rigidity, and their connections to topology.