Spectra of elements in the group ring of SU(2)

Gamburd, Alex, Jakobson, Dmitry, and Sarnak, Peter. "Spectra of elements in the group ring of SU(2)." Journal of the European Mathematical Society 001.1 (1999): 51-85. <http://eudml.org/doc/277567>.

@article{Gamburd1999,
abstract = {We present a new method for establishing the ‘‘gap” property for finitely generated subgroups of $\operatorname\{SU\}(2)$, providing an elementary solution of Ruziewicz problem on $S^2$ as well as giving many new examples of finitely generated subgroups of $\operatorname\{SU\}(2)$ with an explicit gap. The distribution of the eigenvalues of the elements of the group ring $\mathbf \{R\}[\operatorname\{SU\}(2)]$ in the $N$-th irreducible representation of $\operatorname\{SU\}(2)$ is also studied. Numerical experiments indicate that for a generic (in measure) element of $\mathbf \{R\}[\operatorname\{SU\}(2)]$, the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for $N$ even) and the GSE spacing law (for $N$ odd) as $N\rightarrow \infty $; we establish several results in this direction. For certain special “arithmetic” (or Ramanujan) elements of $\mathbf \{R\}[\operatorname\{SU\}(2)]$ the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction.},
author = {Gamburd, Alex, Jakobson, Dmitry, Sarnak, Peter},
journal = {Journal of the European Mathematical Society},
keywords = {finitely generated subgroups of $\operatorname\{SU\}(2)$; GOE spacing law; GSE spacing law; unfolded consecutive spacing distribution; Ruziewicz problem; finitely additive measure},
language = {eng},
number = {1},
pages = {51-85},
publisher = {European Mathematical Society Publishing House},
title = {Spectra of elements in the group ring of SU(2)},
url = {http://eudml.org/doc/277567},
volume = {001},
year = {1999},
}

TY - JOUR
AU - Gamburd, Alex
AU - Jakobson, Dmitry
AU - Sarnak, Peter
TI - Spectra of elements in the group ring of SU(2)
JO - Journal of the European Mathematical Society
PY - 1999
PB - European Mathematical Society Publishing House
VL - 001
IS - 1
SP - 51
EP - 85
AB - We present a new method for establishing the ‘‘gap” property for finitely generated subgroups of $\operatorname{SU}(2)$, providing an elementary solution of Ruziewicz problem on $S^2$ as well as giving many new examples of finitely generated subgroups of $\operatorname{SU}(2)$ with an explicit gap. The distribution of the eigenvalues of the elements of the group ring $\mathbf {R}[\operatorname{SU}(2)]$ in the $N$-th irreducible representation of $\operatorname{SU}(2)$ is also studied. Numerical experiments indicate that for a generic (in measure) element of $\mathbf {R}[\operatorname{SU}(2)]$, the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for $N$ even) and the GSE spacing law (for $N$ odd) as $N\rightarrow \infty $; we establish several results in this direction. For certain special “arithmetic” (or Ramanujan) elements of $\mathbf {R}[\operatorname{SU}(2)]$ the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction.
LA - eng
KW - finitely generated subgroups of $\operatorname{SU}(2)$; GOE spacing law; GSE spacing law; unfolded consecutive spacing distribution; Ruziewicz problem; finitely additive measure
UR - http://eudml.org/doc/277567
ER -