On growth of cocycles of isometric representations on $L^p$-spaces
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| Format: | Preprint |
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2025
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| author | Neumann, Antonio López Paucar, Juan |
| author_facet | Neumann, Antonio López Paucar, Juan |
| contents | We study different notions of asymptotic growth for 1-cocycles of isometric representations on Banach spaces. One can see this as a way of quantifying the absence of fixed point properties on such spaces. Inspired by the work of Lafforgue, we show the following dichotomy: for a compactly generated group $G$, either all 1-cocycles of $G$ taking values in $L^p$-spaces are bounded (this is Property $FL^p$) or there exists a 1-cocycle of $G$ taking values in an $L^p$-space with relatively fast growth. We also obtain upper and lower bounds on the average growth of harmonic 1-cocycles with values in Banach spaces with convexity properties. As a consequence, we obtain bounds on the average growth of all 1-cocycles with values in $L^p$-spaces for groups with property $(T)$. Lastly, we show that for a compactly generated group $G$, the existence of a 1-cocycle with compression larger than $\sqrt{n}$ implies the Liouville property for a large family of probability measures on $G$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_12808 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On growth of cocycles of isometric representations on $L^p$-spaces Neumann, Antonio López Paucar, Juan Group Theory Functional Analysis 20J06, 22D05, 22D55, 46B08 We study different notions of asymptotic growth for 1-cocycles of isometric representations on Banach spaces. One can see this as a way of quantifying the absence of fixed point properties on such spaces. Inspired by the work of Lafforgue, we show the following dichotomy: for a compactly generated group $G$, either all 1-cocycles of $G$ taking values in $L^p$-spaces are bounded (this is Property $FL^p$) or there exists a 1-cocycle of $G$ taking values in an $L^p$-space with relatively fast growth. We also obtain upper and lower bounds on the average growth of harmonic 1-cocycles with values in Banach spaces with convexity properties. As a consequence, we obtain bounds on the average growth of all 1-cocycles with values in $L^p$-spaces for groups with property $(T)$. Lastly, we show that for a compactly generated group $G$, the existence of a 1-cocycle with compression larger than $\sqrt{n}$ implies the Liouville property for a large family of probability measures on $G$. |
| title | On growth of cocycles of isometric representations on $L^p$-spaces |
| topic | Group Theory Functional Analysis 20J06, 22D05, 22D55, 46B08 |
| url | https://arxiv.org/abs/2501.12808 |