No countable family of bounded-degree graphs admitting finitely cobounded coarse quasi-actions contains every finitely generated group as a coarse embedding, resolving conjectures on the non-existence of universal Cayley graphs and quasi-isometry classes.
Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces
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abstract
In this article it is proved, that every locally compact second countable group has a left invariant metric d, which generates the topology on G, and which is proper, ie. every closed d-bounded set in G is compact. Moreover, we obtain the following extension of a result due to N. Brown and E. Guentner: Every locally compact second countable $G$ admits a proper affine action on the reflexive and strictly convex Banach space $\bigoplus^{\infty}_{n=1} L^{2n}(G, d\mu),$ where the direct sum is taken in the $l^2$-sense.
years
2026 2verdicts
UNVERDICTED 2representative citing papers
Discrete groups with finite-complexity coarse embeddings into ⊕_p ℓ^{2p}(N) satisfy the strong Novikov conjecture.
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Obstructions to coarse universality for finitely generated groups
No countable family of bounded-degree graphs admitting finitely cobounded coarse quasi-actions contains every finitely generated group as a coarse embedding, resolving conjectures on the non-existence of universal Cayley graphs and quasi-isometry classes.
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Embedding complexity into Banach spaces and the strong Novikov conjecture
Discrete groups with finite-complexity coarse embeddings into ⊕_p ℓ^{2p}(N) satisfy the strong Novikov conjecture.