scholarly journals Finite presentations for spherical/braid twist groups from decorated marked surfaces

2020 ◽  
Vol 13 (2) ◽  
pp. 501-538
Author(s):  
Yu Qiu ◽  
Yu Zhou
Keyword(s):  
Finite Presentations

scholarly journals Pro-finite Presentations

2001 ◽  
Vol 242 (2) ◽  
pp. 672-690 ◽  
Author(s):  
Alexander Lubotzky
Keyword(s):  
Finite Presentations

scholarly journals Generalized small cancellation presentations for automatic groups

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert H. Gilman
Keyword(s):  
Small Cancellation ◽  
Automatic Groups ◽  
Finite Presentations

AbstractBy a result of Gersten and Short finite presentations satisfying the usual non-metric small cancellation conditions present biautomatic groups. We show that in the case in which all pieces have length 1, a generalization of the C(3)-T(6) condition yields a larger collection of biautomatic groups.

scholarly journals PRESENTATIONS OF AFFINE KAC–MOODY GROUPS

2018 ◽  
Vol 6 ◽  
Author(s):  
INNA CAPDEBOSCQ ◽  
KARINA KIRKINA ◽  
DMITRIY RUMYNIN
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Chevalley Groups ◽  
Generators And Relations ◽  
Simply Connected ◽  
Finite Presentations

How many generators and relations does $\text{SL}\,_{n}(\mathbb{F}_{q}[t,t^{-1}])$ need? In this paper we exhibit its explicit presentation with $9$ generators and $44$ relations. We investigate presentations of affine Kac–Moody groups over finite fields. Our goal is to derive finite presentations, independent of the field and with as few generators and relations as we can achieve. It turns out that any simply connected affine Kac–Moody group over a finite field has a presentation with at most 11 generators and 70 relations. We describe these presentations explicitly type by type. As a consequence, we derive explicit presentations of Chevalley groups $G(\mathbb{F}_{q}[t,t^{-1}])$ and explicit profinite presentations of profinite Chevalley groups $G(\mathbb{F}_{q}[[t]])$.

scholarly journals Metabelian groups: Full-rank presentations, randomness and Diophantine problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Albert Garreta ◽  
Leire Legarreta ◽  
Alexei Miasnikov ◽  
Denis Ovchinnikov
Keyword(s):  
Normal Subgroup ◽  
Full Rank ◽  
Direct Decomposition ◽  
Metabelian Groups ◽  
Abelian Normal Subgroup ◽  
Diophantine Problem ◽  
Diophantine Problems ◽  
Finitely Presented ◽  
Finite Presentations

AbstractWe study metabelian groups 𝐺 given by full rank finite presentations \langle A\mid R\rangle_{\mathcal{M}} in the variety ℳ of metabelian groups. We prove that 𝐺 is a product of a free metabelian subgroup of rank \max\{0,\lvert A\rvert-\lvert R\rvert\} and a virtually abelian normal subgroup, and that if \lvert R\rvert\leq\lvert A\rvert-2, then the Diophantine problem of 𝐺 is undecidable, while it is decidable if \lvert R\rvert\geq\lvert A\rvert. We further prove that if \lvert R\rvert\leq\lvert A\rvert-1, then, in any direct decomposition of 𝐺, all factors, except one, are virtually abelian. Since finite presentations have full rank asymptotically almost surely, metabelian groups finitely presented in the variety of metabelian groups satisfy all the aforementioned properties asymptotically almost surely.

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