Topic
Boundedly generated group
About: Boundedly generated group is a research topic. Over the lifetime, 4 publications have been published within this topic receiving 30 citations.
Papers
TL;DR: In this article, it was shown that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and gave sublinear upper bounds on the ranks of arbitrary finite indices subgroups of such groups.
Abstract: We show that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and give (sublinear) upper bounds on the ranks of arbitrary finite index subgroups of boundedly generated groups (examples which come close to achieving these bounds are presented). This proves a strong form of a conjecture of Abert, Jaikin-Zapirain, and Nikolov which asserts that the rank gradient of infinite boundedly generated residually finite groups is $0$. Furthermore, our first result establishes a variant of a conjecture of Lubotzky on the ranks of finite index subgroups of special linear groups over the integers, and is analogous to a result of Pyber and Segal for solvable groups.
2 citations
TL;DR: In this paper, it was shown that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and gave sublinear upper bounds on the ranks of arbitrary finite indices subgroups of such groups.
Abstract: We show that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and give (sublinear) upper bounds on the ranks of arbitrary finite index subgroups of boundedly generated groups (examples which come close to achieving these bounds are presented). This proves a strong form of a conjecture of Abert, Jaikin-Zapirain, and Nikolov which asserts that the rank gradient of infinite boundedly generated residually finite groups is $0$. Furthermore, our first result establishes a variant of a conjecture of Lubotzky on the ranks of finite index subgroups of special linear groups over the integers, and is analogous to a result of Pyber and Segal for solvable groups.
TL;DR: The existence of infinite simple boundedly simple 2-generated groups with a free noncyclic subgroup was proved in this paper, and the existence of an infinite simple 3-generated 2-group with the same subgroup is also proved.
Abstract: The existence of an infinite simple boundedly generated 2-generated group and the existence of a boundedly simple 2-generated group containing a free noncyclic subgroup are proved.
TL;DR: In this paper, a counterexample is constructed as a group presented by generators and defining relations, where every element of G has a unique presentation in the form a k 1 1 1...a kn n where k i ∈ Z.
Abstract: The following question was asked by V. V. Bludov in The Kourovka Notebook in 1995: If a torsion-free group G has a finite system of generators a 1 ,..., a n such that every element of G has a unique presentation in the form a k 1 1 ...a kn n where k i ∈ Z, is it true that G is virtually polycyclic? The answer is "not always." A counterexample is constructed in this paper as a group presented by generators and defining relations.
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2016 | 1 |
| 2015 | 1 |
| 2007 | 1 |
| 2005 | 1 |