Alberto Takase


UC Irvine


Tuesday, October 26, 2021 - 1:00pm to 2:00pm



Suppose that at each vertex of the Cayley graph of a finitely generated group G is a person holding a dollar. Everybody is told to pass their dollar bill to a neighbor. This can be done so that each person’s net worth increases if and only if the group G is non-amenable. Thus, one can think of non-amenable groups as those where Ponzi schemes can benefit everyone. The Cayley graph of the free group with two generators is an infinite 4-valent tree. If everyone passes their dollar towards the origin then everyone’s net worth increases! Because we live in a world where Ponzi schemes don't work, we restrict our attention to amenable groups such as the integer lattice. For dynamically-defined operator families, the Hausdorff distance of the spectra is estimated by the distance of the underlying dynamical systems while the group is amenable. We prove that if the group has strict polynomial growth and both the group action and the coefficients are Lipschitz continuous, then the spectral estimate has a square root behavior or, equivalently, the spectrum map is $ \frac{1}{2} $-Hölder continuous.