Speaker: Oren Becker

Title: Locally testable groups

Abstract:

Arzhantseva and Paunescu [AP2015] showed that if two permutations X and Y in Sym(n) nearly commute (i.e. XY is close to YX), then the pair (X,Y) is close to a pair of permutations that really commute.

We regard this result as a property of the equation ab=ba and say that this equation is "locally testable" (aka stable). The general question is: which equations (or sets of equations) are locally testable. One sees that the answer depends only on the abstract group defined by the equations (as relations on the generators) rather than on the equations themselves. This leads to the notion of "locally testable groups" (aka "stable groups").

So, the main result of [AP] is that that abelian groups are locally testable, while Glebsky and Rivera [GR2009] showed that the same is true for every finite group. In addition, [GR] show that a locally testable sofic group must be residually finite.

In a joint work with Alex Lubotzky (work in progress), we develop an ensemble of tools to check whether or not a given group is locally testable. As an application, we can answer two questions of [AP] by showing that the Baumslag-Solitar group BS(1,2) is locally testable and that there exists a finitely presented solvable (hence amenable) residually finite group which is not locally testable. In addition, we prove that sofic groups with property (T) are not locally testable, and that the discrete Heisenberg group is locally testable.

Title: Locally testable groups

Abstract:

Arzhantseva and Paunescu [AP2015] showed that if two permutations X and Y in Sym(n) nearly commute (i.e. XY is close to YX), then the pair (X,Y) is close to a pair of permutations that really commute.

We regard this result as a property of the equation ab=ba and say that this equation is "locally testable" (aka stable). The general question is: which equations (or sets of equations) are locally testable. One sees that the answer depends only on the abstract group defined by the equations (as relations on the generators) rather than on the equations themselves. This leads to the notion of "locally testable groups" (aka "stable groups").

So, the main result of [AP] is that that abelian groups are locally testable, while Glebsky and Rivera [GR2009] showed that the same is true for every finite group. In addition, [GR] show that a locally testable sofic group must be residually finite.

In a joint work with Alex Lubotzky (work in progress), we develop an ensemble of tools to check whether or not a given group is locally testable. As an application, we can answer two questions of [AP] by showing that the Baumslag-Solitar group BS(1,2) is locally testable and that there exists a finitely presented solvable (hence amenable) residually finite group which is not locally testable. In addition, we prove that sofic groups with property (T) are not locally testable, and that the discrete Heisenberg group is locally testable.

## Date:

Thu, 24/11/2016 - 10:30 to 11:30

## Location:

Ross 70