Groups with positive rank gradient and their actions

2016/01/30 by Mark Shusterman, Shusterman, Mark
Mathematics · #20E06 #20E07 #20E18 #20E26 #20F05 #20F65 #20F69 #43A05 #58E40 #60B15 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #math.CO #math.GR #msc:20E06 #msc:20E07 #msc:20E18 #msc:20E26 #msc:20F05 #msc:20F65 #msc:20F69 #msc:43A05 #msc:58E40 #msc:60B15

paper · pdf · doi:10.48550/arxiv.1602.00144

arxiv created 2016/01/30 · arxiv updated 2016/02/02

Abstract

We show that given a finitely generated LERF group G with positive rank gradient, and finitely generated subgroups A,B ≤ G of infinite index, one can find a finite index subgroup B₀ of B such that [G : ⟨ A ∪ B₀ ⟩] = ∞. This generalizes a theorem of Olshanskii on free groups. We conclude that a finite product of finitely generated subgroups of infinite index does not cover G. We construct a transitive virtually faithful action of G such that the orbits of finitely generated subgroups of infinite index are finite. Some of the results extend to profinite groups with positive rank gradient.

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