LERF, tameness and Simon’s conjecture

Simon’s conjecture for fibered knots

We prove Simon’s conjecture for fibered knots in S: a fibered knot group surjects at most finitely many other knot groups.

Tameness of Hyperbolic 3-manifolds

Marden conjectured that a hyperbolic 3-manifold M with finitely generated fundamental group is tame, i.e. it is homeomorphic to the interior of a compact manifold with boundary [42]. Since then, many consequences of this conjecture have been developed by Kleinian group theorists and 3-manifold topologists. We prove this conjecture in theorem 10.2, actually in slightly more generality for PNC ma...

Tameness from Large Cardinal Axioms

We show that Shelah’s Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ tame and applying the categoricity transfer of Grossberg and VanDieren [GV06a]. These techniques also apply t...

Presentation Length and Simon’s Conjecture

The Bender method in groups of finite Morley rank

The algebraicity conjecture for simple groups of finite Morley rank, also known as the Cherlin-Zilber conjecture, states that simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields. In the last 15 years, the main line of attack on this problem has been Borovik’s program of transferring methods from finite group theory. Borovik’s program has led to consi...