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The concept of coarse embedding was introduced by Gromov in 1993. It
plays a crucial role in the study of large-scale geometry of groups
and the Novikov higher signature conjecture. Coarse amenability, also
known as Guoliang Yu’s property A, is a weak amenability-type
condition that is satisfied by many known metric spaces. It implies
the existence of a coarse embedding into a Hilbert space. In this
expository talk, we discuss the interplay between infinite expander
graphs, coarse amenability, and coarse embeddings. We present several
’monster’ constructions, in the setting of metric spaces of bounded
geometry, including a recent construction, jointly with Romain
Tessera, of relative expander graphs which do not weakly contain any
expander.
This research was partially supported by my ERC grant ANALYTIC no. 259527.
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