arXiv subjects

Abdelrhman Elkasapy

Publications and source records attributed to Abdelrhman Elkasapy.

3 recordsLinked to original sources

arXiv2016

A new construction for the shortest non-trivial element in the lower central series

We provide a new upper bound for the length for the shortest non-trivial element in the lower central series $γ_n(\mathbb{F}_2)$ of the free group on two generators. We prove that it has an asymptotic behaviour of the form $O(n^{\log_φ(2)})$, where $φ=1.618...$ is the golden ratio. This new technique is used to provide new estimates on the length of laws for finite groups and on almost laws for compact groups.

math.GR↗

arXiv2013

On the length of the shortest non-trivial element in the derived and the lower central series

We provide upper and lower bounds on the length of the shortest non-trivial element in the derived series and lower central series in the free group on two generators. The techniques are used to provide new estimates on the nilpotent residual finiteness growth and on almost laws for compact groups.

math.GR↗

arXiv2012

About Goto's method showing surjectivity of word maps

Let F be the free group on two letters. For ω \in F we study the associated word map ω: SU(n) \times SU(n) \rightarrow SU(n). Extending a method of Goto, we show that for ω not in the second derived subgroup F^(2) of F, there are infinitely many n \in N such that the associated word map ω : SU(n) \times SU(n) \rightarrow SU(n) is surjective.

math.GR↗