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On the Tits alternative for a class of finitely presented groups with a special focus on symbolic computations. (English) Zbl 1392.20024

Kahrobaei, Delaram (ed.) et al., Algebra and computer science. Joint AMS-EMS-SPM meeting algebra and computer science, Porto, Portugal, June 10–13, 2015. Joint mathematics meetings groups, algorithms, and cryptography, San Antonio, TX, USA, January 10–13, 2015. Joint AMS-Israel Mathematical Union meeting applications of algebra to cryptography, Tel-Aviv, Israel, June 16–19, 2014. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2303-2/pbk; 978-1-4704-3587-5/ebook). Contemporary Mathematics 677, 145-169 (2016).
Summary: A group is said to satisfy the Tits alternative if it contains a non-abelian free subgroup or is virtually solvable, that is, it contains a solvable subgroup of finite index. This property was established by J. Tits, who proved that it is satisfied by finitely generated linear groups over a field [J. Algebra 20, 250–270 (1972; Zbl 0236.20032)].
In this paper, we consider groups admitting a presentation
\[ \langle a,b,c\mid a^{l}=b^{m}=c^{n}=R_{1}^{p}\left (a,b\right )=R_{2}^{q}\left (a,c\right )=\left (abc\right )^{r}=1 , \rangle \]
\(2\leq l,m,n,p,q,r\), where each \(R_{i}\left (x,y\right )\) is a cyclically reduced word involving both \(x\) and \(y\). Many of these groups appear in different contexts, inter alia, as fundamental groups of hyperbolic orbifolds or as factor groups of the extended modular group. To prove the Tits alternative for these groups under certain conditions concerning their presentations we build on previous work on the Tits alternative for generalized tetrahedron groups and apply Gröbner bases computations in non-commutative polynomial rings. Furthermore, the groups with \(R_1\left (a,b\right )=a^\alpha b^\beta \) and \(R_2\left (a,c\right )~=~a^\gamma c^\delta \), \(1\leq \alpha \), \(\gamma <l\), \(1\leq \beta <m\), \(1\leq \delta <n\) are proved to satisfy the Tits alternative and a classification of the finite groups among these groups is presented. Additionally, we remark in which cases these groups are hyperbolic.
For the entire collection see [Zbl 1357.00047].

MSC:

20F05 Generators, relations, and presentations of groups
20E07 Subgroup theorems; subgroup growth
20F67 Hyperbolic groups and nonpositively curved groups
20H20 Other matrix groups over fields

Citations:

Zbl 0236.20032

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ApCoCoA
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