Groups, Geometry, and Dynamics

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Volume 9, Issue 1, 2015, pp. 237–273
DOI: 10.4171/GGD/312

Published online: 2015-04-28

Relative subgroup growth and subgroup distortion

Tara C. Davis[1] and Alexander Yu. Olshanskii[2]

(1) Hawaii Pacific University, Honolulu, USA
(2) Vanderbilt University, Nashville, United States

We study the relative growth of finitely generated subgroups in finitely generated groups, and the corresponding distortion function of the embeddings. We explore which functions are equivalent to the relative growth functions and distortion functions of finitely generated subgroups. We also study the connections between these two asymptotic invariants of group embeddings. We answer Gromov’s question on the relationship between distortion and connectivity radius functions. We give conditions under which a length function on a finitely generated group can be extended to a length function on a larger group.

Keywords: Membership problem, embeddings of groups, length function, distortion, relative growth

Davis Tara, Olshanskii Alexander: Relative subgroup growth and subgroup distortion. Groups Geom. Dyn. 9 (2015), 237-273. doi: 10.4171/GGD/312