Expanders and box spaces

Box spaces of finitely generated groups are disjoint union of Cayley graphs of finite quotients associated with some decreasing sequence of finite index normal subgroups of the given group. In 1973 Margulis gave the first explicit construction of expanders by proving that box spaces of property (T) groups are expanders. In 2012 Mendel and Naor showed the existence of two expanders $F_1,F_2$ such that $F_1$ does not coarsely embed into $F_2$. In February 2015, Hume constructed a continuum of expanders with unbounded girth, not coarsely embedding into one another. In joint work with Ana Khukhro, we construct countably many expanders with bounded girth, as box spaces of groups with property $(\tau)$, and prove that they do not coarsely embed into one another.

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