Timeline for Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| May 27, 2014 at 5:18 | comment | added | Venkataramana | @David Loeffler: the natural cases where CSP is not known: co-compact (arithmetic) lattices in $SL_3({\mathbb R})$ | |
| May 10, 2013 at 6:28 | comment | added | David Loeffler | I am no expert here, but I thought the CSP was still open for lots of real rank 1 groups. I guess it all depends wht kind of groups you consider "natural". | |
| May 8, 2013 at 20:50 | comment | added | Misha | See also mathoverflow.net/questions/118376 for more on low index example and why index 7 is the least possible. | |
| May 8, 2013 at 20:20 | comment | added | Jim Humphreys | @David: A helpful answer, including the clarification about passage to an algebraic group and arithmetic subgroups in the formulation. As Raghunathan's 2004 bibliography shows, the CSP has been solved in the bulk of natural cases where it comes up (especially in his work with Prasad); so it seems to be an overstatement to call the problem "far from solved". Maybe I'm misunderstanding what are the "more general reductive groups" you mention. | |
| May 8, 2013 at 20:18 | comment | added | ACL | To construct non-congruence subgroups of $PSL(2,\mathbf Z)$, it is easier to start from the congruence subgroup $\Gamma(2)$ — matrices congruent to identity modulo $2$, for this group is free on two generators. Since the symmetric group $S_k$ is generated by a transposition and a $k$-cycle, it is a quotient of $\Gamma(2)$. | |
| May 8, 2013 at 20:06 | comment | added | David Loeffler | I added an explicit example for index 7. | |
| May 8, 2013 at 20:06 | history | edited | David Loeffler | CC BY-SA 3.0 |
Explicit $S_7$ example
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| May 8, 2013 at 20:05 | comment | added | François Brunault | Another useful reference are K. Conrad's notes : math.uconn.edu/~kconrad/blurbs/grouptheory/SL(2,Z).pdf | |
| May 8, 2013 at 20:00 | history | edited | David Loeffler | CC BY-SA 3.0 |
Corrected a mistake about index 7
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| May 8, 2013 at 8:24 | vote | accept | Marc Palm | ||
| May 8, 2013 at 8:18 | history | answered | David Loeffler | CC BY-SA 3.0 |