Searcharxiv⌕ Search

arXiv subjects

Bharatram Rangarajan

Publications and source records attributed to Bharatram Rangarajan.

3 recordsLinked to original sources
arXiv2023

Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups

It is, by now, classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list: uniform stability with respect to the family of unitary operators on finite-dimensional Hilbert spaces equipped with submultiplicative norms. Namely, we show that for (most) high-rank lattices, every finite-dimensional unitary "almost-representation" of $Γ$ is a small deformation of a (true) unitary representation. This extends a result of Kazhdan (1983) for amenable groups and of Burger-Ozawa-Thom (2013) for SL(n,Z) (for n>2). Towards this goal, we first build an elaborate cohomological theory capturing the obstruction to such stability, and show that the vanishing of second cohomology implies uniform stability in this setting. This cohomology can be roughly thought of as an asymptotic version of bounded cohomology, and sheds light on a question raised in Monod (2006) about a possible connection between vanishing of second bounded cohomology and Ulam stability.

math.GR↗
arXiv2023

Ulam stability of lamplighters and Thompson groups

We show that a large family of groups is uniformly stable relative to unitary groups equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten $p$-norms. These include lamplighters $Γ\wr Λ$ where $Λ$ is infinite and amenable, as well as several groups of dynamical origin such as the classical Thompson groups $F, F', T$ and $V$. We prove this by means of vanishing results in asymptotic cohomology, a theory introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable for studying uniform stability. Along the way, we prove some foundational results in asymptotic cohomology, and use them to prove some hereditary features of Ulam stability. We further discuss metric approximation properties of such groups, taking values in unitary or symmetric groups.

math.GR↗