On Ulam stability

By M. Burger, Nathanial P. Brown Narutaka Ozawa, Andreas Thom. 2010

102 citations recorded by the source corpus.

We study $\epsilon$-representations of discrete groups by unitary operators on a Hilbert space. We define the notion of Ulam stability of a group which loosely means that finite-dimensional $\epsilon$-represendations are uniformly close to unitary representations. One of our main results is that certain lattices in connected semi-simple Lie groups of higher rank are Ulam stable. For infinite-dimensional $\epsilon$-representations, the similarly defined notion of strong Ulam stability is defined and it is shown that groups with free subgroups are not strongly Ulam stable. We also study deformation rigidity of unitary representations and show that groups containing a free subgroup are not deformation rigid.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0a7f1712d24b0b348de68b96a5d861958457af88

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In Algebra and Number Theory

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Character rigidity for special linear groups (2013, result): “t the following holds. For every inﬁnite ﬁeld k, n∈ Nand every δ-homomorphism ϕ: PSLm(k) → U(n), we have k1− ϕ(g)k2 ≤ εfor all g∈ PSLm(k). This result should be compared with some stronger results in [6] which apply to groups SLn(BS−1) for n≥ 3 and measure almost homomorphisms in the operator norm i”
STABILITY, COHOMOLOGY VANISHING, AND NONAPPROXIMABLE GROUPS (2017, background): “Local, since we are only interested in the behavior of φn on the set of relations R (compare with the uniform situation [14]), and discrete, because the family of homomorphisms is indexed by the natural numbers.”
FINITARY APPROXIMATIONS OF GROUPS AND THEIR APPLICATIONS (2017, result): “relations have to be considered at a time. This is in stark contrast to the uniform situation, which – starting with the work of Grove-Karcher-Ruh and Kazhdan [26,34] – is much better understood, see [7,13]. Again, there are longstanding problems that ask if any group exists which is not approximate”
Stability of group relations under small Hilbert–Schmidt perturbations (2017, background): “One should distinguish ǫ-almost homomorphisms from completely different notions of group quasi-representations and δ-homomorphisms as in [28, 6], where almost multiplicativity is required on the whole group.”