Groups with positive rank gradient and their actions

Year, pages: 2018, 353 - 360

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 $[G : \langle A \cup B₀ \rangle] = ∞$. 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.

Shusterman, M. 2018. Groups with positive rank gradient and their actions. In Mathematica Slovaca, vol. 68, no.2, pp. 353-360. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0106

Shusterman, M. (2018). Groups with positive rank gradient and their actions. Mathematica Slovaca, 68(2), 353-360. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0106