Abstract: 

We study quasi-isometric embeddings from a random Bernoulli percolation sample on the product of two regular trees into the product itself, and show some properties that can be seen as extensions of quasi-isometric rigidity of higher rank non-uniform lattices. We also show that two independent samples are almost surely not quasi-isometric equivalent, thus confirming that such a phenomenon occurs in the higher-rank setting, as conjectured by Abert. Joint work with Zhiqiang Li and Ranfeng Yu.

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