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Representations of groups with CAT(0) fixed point property

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We show that certain representations over fields with positive characteristic of groups having CAT\((0)\) fixed point property \(\mathrm{F}\mathcal {B}_{\widetilde{A}_n}\) have finite image. In particular, we obtain rigidity results for representations of the following groups: the special linear group over \({\mathbb {Z}}\), \({\mathrm{SL}}_k({\mathbb {Z}})\), the special automorphism group of a free group, \(\mathrm{SAut}(F_k)\), the mapping class group of a closed orientable surface, \(\mathrm{Mod}(\Sigma _g)\), and many other groups. In the case of characteristic zero, we show that low dimensional complex representations of groups having CAT\((0)\) fixed point property \(\mathrm{F}\mathcal {B}_{\widetilde{A}_n}\) have finite image if they always have compact closure.

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The author would like to thank the referee for many helpful comments.

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Correspondence to Olga Varghese.

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Research partially supported by SFB 878.

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Varghese, O. Representations of groups with CAT(0) fixed point property. Arch. Math. 111, 231–238 (2018).

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