# On Ulam stability - Mathematics > Functional Analysis

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Abstract: We study $\epsilon$-representations of discrete groups by unitary operatorson a Hilbert space. We define the notion of Ulam stability of a group whichloosely means that finite-dimensional $\epsilon$-represendations are uniformlyclose to unitary representations. One of our main results is that certainlattices in connected semi-simple Lie groups of higher rank are Ulam stable.For infinite-dimensional $\epsilon$-representations, the similarly definednotion of strong Ulam stability is defined and it is shown that groups withfree subgroups are not strongly Ulam stable. We also study deformation rigidityof unitary representations and show that groups containing a free subgroup arenot deformation rigid.

Autor: Marc Burger, Narutaka Ozawa, Andreas Thom

Fuente: https://arxiv.org/