A Notion of Dimension based on Probability on Groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909183021940736 |
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| author | Georgakopoulos, Agelos |
| author_facet | Georgakopoulos, Agelos |
| contents | We introduce notions of dimension of an infinite group, or more generally, a metric space, defined using percolation. Roughly speaking, the percolation dimension $pdim(G)$ of a group $G$ is the fastest rate of decay of a symmetric probability measure $μ$ on $G$, such that Bernoulli percolation on $G$ with connection probabilities proportional to $μ$ behaves like a Poisson branching process with parameter 1 in a sense made precise below. We show that $pdim(G)$ has several natural properties: it is monotone decreasing with respect to subgroups and quotients, and coincides with the growth rate exponent for several classes of groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_17278 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Notion of Dimension based on Probability on Groups Georgakopoulos, Agelos Probability Combinatorics Group Theory Geometric Topology 60K35, 82B43, 20P05, 20F69, 60D05, 05E18, 05C63 We introduce notions of dimension of an infinite group, or more generally, a metric space, defined using percolation. Roughly speaking, the percolation dimension $pdim(G)$ of a group $G$ is the fastest rate of decay of a symmetric probability measure $μ$ on $G$, such that Bernoulli percolation on $G$ with connection probabilities proportional to $μ$ behaves like a Poisson branching process with parameter 1 in a sense made precise below. We show that $pdim(G)$ has several natural properties: it is monotone decreasing with respect to subgroups and quotients, and coincides with the growth rate exponent for several classes of groups. |
| title | A Notion of Dimension based on Probability on Groups |
| topic | Probability Combinatorics Group Theory Geometric Topology 60K35, 82B43, 20P05, 20F69, 60D05, 05E18, 05C63 |
| url | https://arxiv.org/abs/2404.17278 |