Concise presentations of direct products
Direct powers of perfect groups admit more concise presentations than one might naively suppose. If H1(G;Z) = H2(G;Z) = 0, then Gn has a presentation with O(log n) generators and O(log n)3 relators. If, in addition, there is an element g 2 G that has infinite order in every non-trivial quotient of G...
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| Médium: | Journal article |
| Jazyk: | English |
| Vydáno: |
American Mathematical Society
2022
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| Shrnutí: | Direct powers of perfect groups admit more concise presentations than one might naively suppose. If H1(G;Z) = H2(G;Z) = 0, then Gn has a presentation with O(log n) generators and O(log n)3 relators. If, in addition, there is an element g 2 G that has infinite order in every non-trivial quotient of G, then Gn has a presentation with d(G) + 1 generators and O(log n) relators. The bounds that we obtain on the deficiency of Gn are not monotone in n; this points to potential counterexamples for the Relation Gap Problem. |
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