Timeline for For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complement is approximated by the growth rate of curvature?
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| Sep 16, 2014 at 16:25 | comment | added | Asaf | one of the definitions of a rank of symmetric space is the rank of the group from which the symmetric space is formed, so indeed, SO(1,n) are rank $1$. For higher rank, Quint proved analogues of PS for Schottky groups in his PhD thesis (which are doable due to the symbolic coding), and I think some results in the general case were proven by Albuquerque. | |
| Sep 16, 2014 at 15:57 | comment | added | Hao Chen | @Asaf, what do you mean by rank 1 spaces? Do you mean pseudo-Euclidean spaces of signature (1,n)? I am indeed interested to know more about any sort of P-S theory in spaces of signature (p,q). | |
| Sep 16, 2014 at 15:55 | vote | accept | Hao Chen | ||
| Sep 14, 2014 at 10:02 | comment | added | Asaf | The dimension=exponent part should be known from some sort of Patterson-Sullivan theory, as some integrability condition for proper Poincare series, especially if you're interested in only rank $1$ spaces. I would check Oh's papers, especially the papers with Mohammadi (If I recall correctly the first counting result was achieved with Kontorovich, and the second one with Mohammadi). McMullen have done some work computing dimensions as well (late 90's or so). | |
| Aug 15, 2014 at 9:34 | answer | added | Hao Chen | timeline score: 1 | |
| Aug 13, 2014 at 15:17 | history | edited | Hao Chen | CC BY-SA 3.0 |
add two labels and minor changes.
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| Jul 10, 2014 at 22:50 | history | asked | Hao Chen | CC BY-SA 3.0 |