Poincaré inequalities, embeddings, and wild groups
By Assaf Naor, Lior Silberman. Compositio Mathematica, 2010
106 citations recorded by the source corpus.
Abstract We present geometric conditions on a metric space (Y,dY) ensuring that, almost surely, any isometric action on Y by Gromov’s expander-based random group has a common fixed point. These geometric conditions involve uniform convexity and the validity of nonlinear Poincaré inequalities, and they are stable under natural operations such as scaling, Gromov–Hausdorff limits, and Cartesian products. We use methods from metric embedding theory to establish the validity of these conditions for a variety of classes of metric spaces, thus establishing new fixed point results for actions of Gromov’s ‘wild groups’.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/4747913822dc48e8db0532575a196efab96b4726
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Nonlinear spectral calculus and super-expanders (2012, background): “In [53, 56] additional results of this type are obtained.”
- Speed of random walks, isoperimetry and compression of finitely generated groups (2015, unknown): “[70] Naor, Assaf; Silberman, Lior, Poincaré inequalities, embeddings, and wild groups, Compos.”
- Comparison of Metric Spectral Gaps (2013, background): “The remaining cases follow from the general fact [83] that if (X, dX) has bounded Nagata dimension then ε(X; δ) > 0 for some δ ∈ (0, 1) (see [58] for more information on Nagata dimension of metric spaces).”
- Assouad's theorem with dimension independent of the snowflaking (2010, background): “When allowing embeddings into infinite dimensional Hilbert space rather than into R , the asymptotics in terms of ε of D(K, ε) in Assouad’s theorem are known [21] (see also [24]).”