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

Alex Gamburd; Dmitry Jakobson; Peter Sarnak

Journal of the European Mathematical Society (1999)

- Volume: 001, Issue: 1, page 51-85
- ISSN: 1435-9855

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topGamburd, 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 -

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