Geodesic
Problem of Intermediate Relative Growth of Subgroups in Solvable and Linear Groups
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems, automata, and infinite groups, Tome 231 (2000), pp. 332-355

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The following problem formulated in [16] is considered: Is it true that the relative growth of any subgroup of a solvable or linear group is either polynomial or exponential? In the general case, a negative answer is obtained for both classes of groups. Certain positive results are also presented. 
@article{TM_2000_231_a12,
     author = {D. V. Osin},
     title = {Problem of {Intermediate} {Relative} {Growth} of {Subgroups} in {Solvable} and {Linear} {Groups}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {332--355},
     publisher = {mathdoc},
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     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2000_231_a12/}
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D. V. Osin. Problem of Intermediate Relative Growth of Subgroups in Solvable and Linear Groups. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems, automata, and infinite groups, Tome 231 (2000), pp. 332-355. http://geodesic.mathdoc.fr/item/TM_2000_231_a12/