Damian Sawicki

Damian Sawicki
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Damian Sawicki

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Mathematics - Metric Geometry (5)
Mathematics - Group Theory (5)
Mathematics - Dynamical Systems (2)
Mathematics - Functional Analysis (2)
Mathematics - Geometric Topology (1)

Publications Authored By Damian Sawicki

For any Banach space $X$, we show that levels of the warped cone over an action of $\Gamma$ on $Y$ are expanders with respect to $X$ if and only if the induced $\Gamma$-representation on $L^2(Y;X)$ has a spectral gap. This gives new examples of graphs that are expanders with respect to all Banach spaces of non-trivial type. Read More

We show that warped cones over actions with spectral gaps do not embed coarsely into large classes of Banach spaces. In particular, there exist warped cones over actions of the free group that do not embed coarsely into $L_p$-spaces and there are warped cones over discrete group actions that do not embed into any Banach space with non-trivial type. Read More

We construct metric spaces that do not have property A yet are coarsely embeddable into the Hilbert space. Our examples are so called warped cones, which were introduced by J. Roe to serve as examples of spaces non-embeddable into a Hilbert space and with or without property A. Read More

The work discusses equivariant asymptotic dimension (also known as "wide equivariant covers", "transfer reducibility", and "$N$-$\mathcal F$-amenability"), a version of asymptotic dimension invented for the proofs of the Farrell-Jones and Borel conjectures. We prove that groups of null equivariant asymptotic dimension are exactly virtually cyclic groups. Moreover, we show that a covering of the boundary always extends to a covering of the whole compactification. Read More

We show for a given metric space $(X,d)$ of asymptotic dimension $n$ that there exists a coarsely and topologically equivalent hyperbolic metric $d'$ of the form $d' = f \circ d$ such that $(X,d')$ is of asymptotic Assouad-Nagata dimension $n$. As a corollary we construct examples of spaces realising strict inequality in the logarithmic law for AN-asdim of a Cartesian product. One of them may be viewed as a counterexample to a specific kind of a Morita-type theorem for AN-asdim. Read More