Extremal behavior of divisibility functions. Bou-Rabee, K. & McReynolds, D. B. *Geometriae Dedicata*, 175(1):407–415, April, 2015. tex.ids= Bou-RabeeMcReynolds2015Extremal publisher: SpringerPaper doi abstract bibtex In this short article, we study the extremal behavior F (n) of divisibility functions D introduced by the ﬁrst author for ﬁnitely generated groups . These functions aim at quantifying residual ﬁniteness and bounds give a measurement of the complexity in verifying a word is non-trivial. We show that ﬁnitely generated subgroups of GL(m, K ) for an inﬁnite ﬁeld K have at most polynomial growth for the function F (n). Consequently, we obtain a dichotomy for the growth rate of log F (n) for ﬁnitely generated subgroups of GL(n, C). We also show that if F (n) log log n, then is ﬁnite. In contrast, when contains an element of inﬁnite order, log n F (n). We end with a brief discussion of some geometric motivation for this work.

@article{bou-rabee_extremal_2015,
title = {Extremal behavior of divisibility functions},
volume = {175},
issn = {0046-5755, 1572-9168},
url = {http://link.springer.com/10.1007/s10711-014-9955-5},
doi = {10.1007/s10711-014-9955-5},
abstract = {In this short article, we study the extremal behavior F (n) of divisibility functions D introduced by the ﬁrst author for ﬁnitely generated groups . These functions aim at quantifying residual ﬁniteness and bounds give a measurement of the complexity in verifying a word is non-trivial. We show that ﬁnitely generated subgroups of GL(m, K ) for an inﬁnite ﬁeld K have at most polynomial growth for the function F (n). Consequently, we obtain a dichotomy for the growth rate of log F (n) for ﬁnitely generated subgroups of GL(n, C). We also show that if F (n) log log n, then is ﬁnite. In contrast, when contains an element of inﬁnite order, log n F (n). We end with a brief discussion of some geometric motivation for this work.},
language = {en},
number = {1},
urldate = {2021-08-11},
journal = {Geometriae Dedicata},
author = {Bou-Rabee, Khalid and McReynolds, D. B.},
month = apr,
year = {2015},
note = {tex.ids= Bou-RabeeMcReynolds2015Extremal
publisher: Springer},
pages = {407--415},
}

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