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Expansion in SL d (𝒪 K /I), I square-free. (English) Zbl 1269.20044
Let S be a fixed symmetric finite subset of SL d (𝒪 K ) that generates a Zariski dense subgroup of SL d (𝒪 K ) when considered as an algebraic group over by restriction of scalars. The author proves that the Cayley graphs of SL d (𝒪 K /I) with respect to the projections of S is an expander family when I ranges over square-free ideals of 𝒪 K if d=2 and K is an arbitrary number field or if d=3 and K=.
20G30Linear algebraic groups over global fields and their integers
20F05Generators, relations, and presentations of groups
20F65Geometric group theory
05C25Graphs and abstract algebra
11B30Arithmetic combinatorics; higher degree uniformity
11B75Combinatorial number theory