Uniform boundedness of $ (SL_2(\mathbb{C}))^{n} $ and $ (PSL_2(\mathbb{C}))^{n} $
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\in G $ can be written as a product of conjugates of elements of $ S $ and their inverses. If $ S\subseteq G $ normall...
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| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2024-11-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241609 |
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| Summary: | 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\in G $ can be written as a product of conjugates of elements of $ S $ and their inverses. If $ S\subseteq G $ normally generates $ G, $ then the length $ \| g\|_{S}\in \mathbb{N} $ of $ g\in G $ with respect to $ S $ is the shortest possible length of a word in $ \text{Conj}_{G}(S^{\pm 1}): = \{h^{-1}sh | h\in G, s\in S \, \text{or} \, s{^{-1}}\in S \} $ expressing $ g. $ We write $ \|G\|_{S} = \text{sup}\{\|g\|_{S} \, |\, \, g\in G\} $ for any normally generating subset $ S $ of $ G. $ The conjugacy diameter of any group $ G $ is $ \Delta(G): = \sup\{ {\|G\|_{S}}\, \, | S\ \text{is a finite normally generating subset of } G \}. $ We say that $ G $ is uniformly bounded if $ \Delta(G) < \infty. $ This concept is a strengthening of boundedness. Motivated by previously known results approximating $ \Delta(G) $ for any algebraic group $ G, $ we find the exact values of the conjugacy diameters of the direct product of finitely many copies of $ SL_2(\mathbb{C}) $ and the direct product of finitely many copies of $ PSL_2(\mathbb{C}). $ We also prove that if $ G_1, \dots, G_n $ be quasisimple groups such that $ G_i $ is uniformly bounded for each $ i\in \{1, \dots, n\}, $ then $ G_1\times\dots \times G_n $ is uniformly bounded. This is also a generalization of some previously known results in the literature. |
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| ISSN: | 2473-6988 |