# On the growth of rank for subgroups of finitely generated groups

## Abstract

In [1] and [2] the functions of rank growth were independently introduced and investigated for subgroups of a finitely generated free group. In the present paper the concept of growth of rank is extended to subgroups of an arbitrary finitely generated group G, and the dependence of the asymptotic behaviour of the above functions on the choice of a finite generating set in G is studied. For a broad class of groups (which includes, in particular, the free polynilpotent groups) estimates for the growth of rank for subgroups are obtained that generalize the wellknown Baumslag-Eidel'kind result on finitely generated normal subgroups. Some problems related to the realization of arbitrary functions as functions of rank growth for subgroups of soluble groups are treated.

- Authors:

- M.V. Lomonosov Moscow State University, Moscow (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 21202879

- Resource Type:
- Journal Article

- Journal Name:
- Sbornik. Mathematics

- Additional Journal Information:
- Journal Volume: 190; Journal Issue: 8; Other Information: DOI: 10.1070/SM1999v190n08ABEH000421; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; FUNCTIONS; GROUP THEORY

### Citation Formats

```
Osin, D V.
```*On the growth of rank for subgroups of finitely generated groups*. United States: N. p., 1999.
Web. doi:10.1070/SM1999V190N08ABEH000421; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Osin, D V.
```*On the growth of rank for subgroups of finitely generated groups*. United States. doi:10.1070/SM1999V190N08ABEH000421; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Osin, D V. Tue .
"On the growth of rank for subgroups of finitely generated groups". United States. doi:10.1070/SM1999V190N08ABEH000421; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
```

```
@article{osti_21202879,
```

title = {On the growth of rank for subgroups of finitely generated groups},

author = {Osin, D V},

abstractNote = {In [1] and [2] the functions of rank growth were independently introduced and investigated for subgroups of a finitely generated free group. In the present paper the concept of growth of rank is extended to subgroups of an arbitrary finitely generated group G, and the dependence of the asymptotic behaviour of the above functions on the choice of a finite generating set in G is studied. For a broad class of groups (which includes, in particular, the free polynilpotent groups) estimates for the growth of rank for subgroups are obtained that generalize the wellknown Baumslag-Eidel'kind result on finitely generated normal subgroups. Some problems related to the realization of arbitrary functions as functions of rank growth for subgroups of soluble groups are treated.},

doi = {10.1070/SM1999V190N08ABEH000421; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},

journal = {Sbornik. Mathematics},

issn = {1064-5616},

number = 8,

volume = 190,

place = {United States},

year = {1999},

month = {8}

}