On finitary approximation properties of groups
1. Equations over groups
Consider equations of the form with unknown and coefficient . We would be happy with solutions in an extension of . Goes back to Bernard Neumann.
Example. cannot be solved over any extension unless abd have the same order. cannot be solved over . So torsion makes it difficult. Also the fact that the exponents sum up to 0 makes it hard.
Definition. Say an equation is non-singular if the sum of exponents is not 0.
Conjecture (Levin). Every nontrivial equation over a torsion free group can be solved.
Conjecture (Kervaire-Laudenbach). Every nonsingular equation can be solved.
Theorem 1 (Klyachko) If is torsion free and the equation is non-singular, then it can be solved.
Theorem 2 (Gerstenhaber-Rothaus) Nonsingular equations over the unitary group can be solved.
The argument is topological (degree theory).
Corollary 3 (Gerstenhaber-Rothaus 1962) Nonsingular equations over finite groups can be solved.
Corollary 4 (Pestov) Nonsingular equations over every group that can be embedded into an abstract quotient of the product of all unitary groups can be solved.
This is the case for every sofic group.
2. Sofic groups
Use the normalized Hamming distance on permutations. A group is sofic if for every finite subset and every , there exists and a map of the group to which is, on , a homomorphism up to metric error .
No group has been shown not to be sofic. This is a hard problem. This is why we study related approximation properties. Before, let us give two more applications of soficity.
2.1. Equations in more variables
Say an equation in several variables is non-singular if setting all coefficients to 1 gives a nontrivial element, called content, of .
Theorem 5 (Klyachko-Thom) If is sofic and the content of the equation does not belong to , then this equation in several variables can be solved in .
We imitate Gerstenhaber-Rothaus’ argument to solve equations over special unitary groups.
2.2. Elek-Szabo’s solution of Kaplansky’s conjecture for sofic groups
They prove that for sofic groups, one-sided inverses in the group’s algebra are two-sided. This holds for finite dimensional cases (matrix algebras)
3. Weakly sofic groups
Same definition, with a class of groups with bi-invariant, conjugation invariant metrics.
Example. Fin=finite groups. Alt=finite alternating groups. PSl=all PSl over finite fileds.
It turns out that weakly sofic groups coincide with Fin-approximable groups, Sofic groups coincide with Alt-approximable groups.
Theorem 6 (Nikolov-Schneider-Thom) Every Fin-approximable grouop has a a nontrivial PSl-approximable quotient.
This relies on Nikolov’s structure theorem on commutators in finite simple groups.
3.1. Link to metric ultraproducts
-approximability is equivalent to embeddability in an ultraproduct of all elements of class .
With John Wilson, we show that metric ultraproducts of and remember their origin.
3.2. Topological groups
These are topological subgroups of metric ultraproducts of elements of class .
4. Sol-approximable abstract groups
Theorem 7 (Nikolov-Schneider-Thom) A nontrivial finitely generated perfect group is not Sol-approximable.
Use the pro--topology on a free group such that . Show that the closure of any product of conjugacy classes of elements of is contained in . Since is perfect, any belings to mod commutators. An element of the commutator subgroup is a product of commutators times an elemet of the intersection of subgroups of the central series.
Theorem 8 (Nikolov-Schneider-Thom) A connected Lie group is Fin-approximable iff it is abelian.
Theorem 9 (Nikolov-Schneider-Thom, Glebsky) is not approximable by any class of finite groups.
We use Nikolov-Segal’s result that if is a finite group and is normal, then is contained in a product of commutators of with elements of a generating set, raised to a power that depends on the rank only.