Semisimple Lie algebras are far from flexibly C-stable, semisimple Lie groups and higher-rank lattices are not strictly C-stable, and free groups are not uniformly flexibly F-stable over any field.
Quasi-morphisms on Free Groups
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abstract
Let F be the free group over a set of two or more generators. R. Brooks constructed an infinite family of quasi-morphisms on F such that an infinite subfamily gives rise to independent classes in the second bounded cohomology of F, which proves that this space is infinite dimensional. We give a simpler proof of this fact using a different type of quasi-morphisms. After computing the Gromov norm of the corresponding bounded classes, we generalize our example to obtain quasi-morphisms on free products, as well as quasi-morphisms into groups without small subgroups, also known as epsilon-representations.
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2024 1verdicts
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Uniform rank metric stability of Lie algebras and groups
Semisimple Lie algebras are far from flexibly C-stable, semisimple Lie groups and higher-rank lattices are not strictly C-stable, and free groups are not uniformly flexibly F-stable over any field.