## Free products in the unit group of the integral group ring of a finite group

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- by Geoffrey Janssens, Eric Jespers and Doryan Temmerman PDF
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## Abstract:

Let $G$ be a finite group and let $p$ be a prime. We continue the search for generic constructions of free products and free monoids in the unit group $\mathcal {U}(\mathbb {Z}G)$ of the integral group ring $\mathbb {Z} G$. For a nilpotent group $G$ with a non-central element $g$ of order $p$, explicit generic constructions are given of two periodic units $b_1$ and $b_2$ in $\mathcal {U}(\mathbb {Z}G)$ such that $\langle b_1, b_2\rangle =\langle b_1\rangle \star \langle b_2 \rangle \cong \mathbb {Z}_p \star \mathbb {Z}_{p}$, a free product of two cyclic groups of prime order. Moreover, if $G$ is nilpotent of class $2$ and $g$ has order $p^n$, then also concrete generators for free products $\mathbb {Z}_{p^k} \star \mathbb {Z}_{p^m}$ are constructed (with $1\leq k,m\leq n$). As an application, for finite nilpotent groups, we obtain earlier results of Marciniak-Sehgal and Gonçalves-Passman. Further, for an arbitrary finite group $G$ we give generic constructions of free monoids in $\mathcal {U}(\mathbb {Z}G)$ that generate an infinite solvable subgroup.## References

- V. Bovdi,
*Free subgroups in group rings*, arXiv:1406.6771, 2014, preprint. - Emmanuel Breuillard,
*On uniform exponential growth for solvable groups*, Pure Appl. Math. Q.**3**(2007), no. 4, Special Issue: In honor of Grigory Margulis., 949–967. MR**2402591**, DOI 10.4310/PAMQ.2007.v3.n4.a4 - Florian Eisele, Ann Kiefer, and Inneke Van Gelder,
*Describing units of integral group rings up to commensurability*, J. Pure Appl. Algebra**219**(2015), no. 7, 2901–2916. MR**3313511**, DOI 10.1016/j.jpaa.2014.09.031 - J. Z. Gonçalves and D. S. Passman,
*Embedding free products in the unit group of an integral group ring*, Arch. Math. (Basel)**82**(2004), no. 2, 97–102. MR**2047662**, DOI 10.1007/s00013-003-4793-y - Jairo Z. Gonçalves and Ángel Del Río,
*A survey on free subgroups in the group of units of group rings*, J. Algebra Appl.**12**(2013), no. 6, 1350004, 28. MR**3063443**, DOI 10.1142/S0219498813500047 - B. Hartley and P. F. Pickel,
*Free subgroups in the unit groups of integral group rings*, Canadian J. Math.**32**(1980), no. 6, 1342–1352. MR**604689**, DOI 10.4153/CJM-1980-104-3 - Martin Hertweck,
*A counterexample to the isomorphism problem for integral group rings*, Ann. of Math. (2)**154**(2001), no. 1, 115–138. MR**1847590**, DOI 10.2307/3062112 *Mini-Workshop: Arithmetik von Gruppenringen*, Oberwolfach Rep.**4**(2007), no. 4, 3209–3239. Abstracts from the mini-workshop held November 25–December 1, 2007; Organized by Eric Jespers, Zbigniew Marciniak, Gabriele Nebe and Wolfgang Kimmerle; Oberwolfach Reports. Vol. 4, no. 4. MR**2463649**, DOI 10.4171/OWR/2007/55- Eric Jespers and Guilherme Leal,
*Generators of large subgroups of the unit group of integral group rings*, Manuscripta Math.**78**(1993), no. 3, 303–315. MR**1206159**, DOI 10.1007/BF02599315 - E. Jespers and Á. del Río,
*Group ring groups. Volume 1: Orders and generic constructions of units*, De Gruyter, Berlin, 2016. - E. Jespers and Á. del Río,
*Group ring groups. Volume 2: Structure theorems of unit groups*, De Gruyter, Berlin, 2016. - Zbigniew S. Marciniak and Sudarshan K. Sehgal,
*Constructing free subgroups of integral group ring units*, Proc. Amer. Math. Soc.**125**(1997), no. 4, 1005–1009. MR**1376998**, DOI 10.1090/S0002-9939-97-03812-4 - D. S. Passman,
*Free products in linear groups*, Proc. Amer. Math. Soc.**132**(2004), no. 1, 37–46. MR**2021246**, DOI 10.1090/S0002-9939-03-07033-3 - Jürgen Ritter and Sudarshan K. Sehgal,
*Construction of units in integral group rings of finite nilpotent groups*, Trans. Amer. Math. Soc.**324**(1991), no. 2, 603–621. MR**987166**, DOI 10.1090/S0002-9947-1991-0987166-9 - Joseph Max Rosenblatt,
*Invariant measures and growth conditions*, Trans. Amer. Math. Soc.**193**(1974), 33–53. MR**342955**, DOI 10.1090/S0002-9947-1974-0342955-9 - S. K. Sehgal,
*Units in integral group rings*, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 69, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. With an appendix by Al Weiss. MR**1242557**

## Additional Information

**Geoffrey Janssens**- Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan $2$, B-1050 Elsene, Belgium
- MR Author ID: 1005538
- Email: Geoffrey.Janssens@vub.be
**Eric Jespers**- Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan $2$, B-1050 Elsene, Belgium
- MR Author ID: 94560
- Email: Eric.Jespers@vub.be
**Doryan Temmerman**- Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan $2$, B-1050 Elsene, Belgium
- Email: Doryan.Temmerman@vub.be
- Received by editor(s): July 8, 2016
- Published electronically: April 4, 2017
- Additional Notes: The first and third authors were supported by Fonds voor Wetenschappelijk Onderzoek (Flanders)

The second author was supported by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Flanders) - Communicated by: Pham Huu Tiep
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 2771-2783 - MSC (2010): Primary 16U60, 20C05, 16S34, 20E06; Secondary 20C10, 20C40
- DOI: https://doi.org/10.1090/proc/13631
- MathSciNet review: 3637929