@article{Aseeri2024, 
author = {Fawaz Aseeri},
title = {Uniform boundedness of  (SL2(C))n and  (PSL2(C))n},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {12},
pages = {33712-33730},
keywords = {normally generating subsets, word norm, conjugacy diameter},
url = {https://www.sciopen.com/article/10.3934/math.20241609},
doi = {10.3934/math.20241609},
abstract = {Let  G be a group and  S be a subset of  G. We say that  S normally generates  G if  G is the normal closure of  S in  G. In this situation, every element  g∈G can be written as a product of conjugates of elements of  S and their inverses. If  S⊆G normally generates  G, then the length  ‖g‖S∈N of  g∈G with respect to  S is the shortest possible length of a word in  ConjG(S±1):={h−1sh|h∈G,s∈Sors−1∈S} expressing  g. We write  ‖G‖S=sup{‖g‖S|g∈G} for any normally generating subset  S of  G. The conjugacy diameter of any group  G is  Δ(G):=sup{‖G‖S|S is a finite normally generating subset of G}. We say that  G is uniformly bounded if  Δ(G)<∞. This concept is a strengthening of boundedness. Motivated by previously known results approximating  Δ(G) for any algebraic group  G, we find the exact values of the conjugacy diameters of the direct product of finitely many copies of  SL2(C) and the direct product of finitely many copies of  PSL2(C). We also prove that if  G1,…,Gn be quasisimple groups such that  Gi is uniformly bounded for each  i∈{1,…,n}, then  G1×⋯×Gn is uniformly bounded. This is also a generalization of some previously known results in the literature.}
}