# ANNALES DE L'INSTITUT FOURIER

An analytic series of irreducible representations of the free group
Annales de l'Institut Fourier, Volume 38 (1988) no. 1, pp. 87-110.

Let ${\mathbf{F}}_{k}$ be a free group on $k$ generators. We construct the series of uniformly bounded representations ${\prod }_{z}$ of ${\mathbf{F}}_{k}$ acting on the common Hilbert space, depending analytically on the complex parameter z, $1/\left(2k-1\right)<|z|<1$, such that each representation ${\prod }_{z}$ is irreducible. If $z$ is real or $|z|=1/\left(\sqrt{2k-1}\right)$ then ${\prod }_{z}$ is unitary; in other cases ${\prod }_{z}$ cannot be made unitary. For $z\ne {z}^{\prime }$ representations ${\prod }_{z}$ and ${\prod }_{{z}^{\prime }}$ are congruent modulo compact operators.

Soit ${\mathbf{F}}_{k}$ un groupe libre avec $k$ générateurs. On construit une série des représentations uniformément bornées ${\prod }_{z}$ de ${\mathbf{F}}_{k}$ qui opèrent sur un espace de Hilbert commun. Les représentations ${\prod }_{z}$ sont irréductibles et dépendent analytiquement d’un paramètre complexe $z$ tel que $1/\left(2k-1\right)<|z|<1$. Pour $z$ réel ou $|z|=1/\left(\sqrt{2k-1}\right)$ les ${\prod }_{z}$ sont unitaires; autrement ${\prod }_{z}$ ne sont pas unitarisables. Pour $z\ne {z}^{\prime }$ les différences ${\prod }_{z}-{\prod }_{{z}^{\prime }}$ sont des opérateurs compacts.

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title = {An analytic series of irreducible representations of the free group},
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Szwarc, Ryszard. An analytic series of irreducible representations of the free group. Annales de l'Institut Fourier, Volume 38 (1988) no. 1, pp. 87-110. doi : 10.5802/aif.1124. https://aif.centre-mersenne.org/articles/10.5802/aif.1124/

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