Timeline for size of smallest generating set of a group
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 27, 2013 at 0:54 | comment | added | Ian Agol | The rank problem is unsolvable among word-hyperbolic groups: ams.org/mathscinet-getitem?mr=1246477 | |
| Sep 27, 2011 at 11:37 | comment | added | user6976 | @Igor: See my update 3. | |
| Sep 27, 2011 at 11:36 | history | edited | user6976 | CC BY-SA 3.0 |
added 220 characters in body
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| Sep 27, 2011 at 11:30 | comment | added | user6976 | @Igor: For $n\ge 28$ you only need 2. See books.google.com/… | |
| Sep 27, 2011 at 10:28 | comment | added | Igor Rivin | Thanks! I am pretty sure that the smallest known generating sets for $SL(n, \mathbb{Z})$ itself has $3$ elements... | |
| Sep 27, 2011 at 9:24 | history | edited | user6976 | CC BY-SA 3.0 |
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| Sep 27, 2011 at 9:23 | comment | added | user6976 | @HW: You are right of course, one just needs to check if $\delta=1, 2,...$. | |
| Sep 27, 2011 at 7:48 | comment | added | HJRW | Mark - Papazoglou's algorithm will find a constant $\delta$ given a presentation of a word-hyperbolic group. | |
| Sep 27, 2011 at 3:28 | comment | added | Andy Putman | I'm not sure what the correct upper bound should be, and would be very interested in knowing the answer. | |
| Sep 27, 2011 at 2:54 | history | edited | user6976 | CC BY-SA 3.0 |
corrected the first part of the answer
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| Sep 27, 2011 at 2:52 | comment | added | user6976 | @Andy: Thanks! Is $n^2-1$ the upper bound for all the lattices in $SL_n$? I will correct the answer. | |
| Sep 27, 2011 at 2:15 | comment | added | Andy Putman | You need more than $2$ generators for lattices in $SL_n(\mathbb{R})$. The abelianizations of the level $p$ congruence subgroups of $SL_n(\mathbb{Z})$ have rank $n^2-1$ (see the paper "On the homology and cohomology of congruence subgroups" by Lee-Szczarba), so you need at least $n^2-1$ generators to generate them. | |
| Sep 26, 2011 at 23:54 | history | answered | user6976 | CC BY-SA 3.0 |