| Abstrakt: |
Expander graphs have been intensively studied in the last four decades (Hoory et al., Bull Am Math Soc, 43(4):439-562, 2006; Lubotzky, Bull Am Math Soc, 49:113-162, 2012). In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every d there are unbounded degree simplicial complexes of dimension d with these properties. However, a major open problem, formulated by Gromov (Geom Funct Anal 20(2):416-526, 2010), is whether bounded degree high dimensional expanders exist for $${d \geq 2}$$ . We present an explicit construction of bounded degree complexes of dimension $${d = 2}$$ which are topological expanders, thus answering Gromov's question in the affirmative. Conditional on a conjecture of Serre on the congruence subgroup property, infinite sub-family of these give also a family of bounded degree coboundary expanders. The main technical tools are new isoperimetric inequalities for Ramanujan Complexes. We prove linear size bounds on $${\mathbb{F}_2}$$ systolic invariants of these complexes, which seem to be the first linear $${\mathbb{F}_2}$$ systolic bounds. The expansion results are deduced from these isoperimetric inequalities. [ABSTRACT FROM AUTHOR] |
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