arXiv subjects

Jan Spakula

Publications and source records attributed to Jan Spakula.

12 recordsLinked to original sources

arXiv2023

Sobolev spaces and uniform boundary representations

We prove uniform boundedness of certain boundary representations on appropriate fractional Sobolev spaces $W^{s,p}$ with $p>1$ for arbitrary Gromov hyperbolic groups. These are closed subspaces of $L^p$ and in particular Hilbert spaces in the case $p=2$. This construction allows us, for an appropriate choice of $p$, to approximate the trivial representation through uniformly bounded representations. This phenomenon does not have analogue in the setting of isometric representations whenever the hyperbolic group considered has the Property (T). The key is the introduction of a notion of metrically conformal operator on a metric space endowed with a conformal structure \`{a} la Mineyev and a metric analogue of the isomorphisms of Sobolev spaces induced by the Cayley transform.

math.GR↗

arXiv2017

Relative commutant pictures of Roe algebras

Let X be a proper metric space, which has finite asymptotic dimension in the sense of Gromov (or more generally, straight finite decomposition complexity of Dranishnikov and Zarichnyi). New descriptions are provided of the Roe algebra of X: (i) it consists exactly of operators which essentially commute with diagonal operators coming from Higson functions (that is, functions on X whose oscillation tends to 0 at infinity) and (ii) it consists exactly of quasi-local operators, that is, ones which have finite epsilon propogation (in the sense of Roe) for every epsilon>0. These descriptions hold both for the usual Roe algebra and for the uniform Roe algebra.

math.OA↗

arXiv2016

Coarse medians and Property A

We prove that uniformly locally finite quasigeodesic coarse median spaces of finite rank and at most exponential growth have Property A. This offers an alternative proof of the fact that mapping class groups have property A.

math.MG↗

arXiv2014

A metric approach to limit operators

We extend the limit operator machinery of Rabinovich, Roch, and Silbermann from $\mathbb{Z}^N$ to (bounded geometry, strongly) discrete metric spaces. We do not assume the presence of any group structure or action on our metric spaces. Using this machinery and recent ideas of Lindner and Seidel, we show that if a metric space X has Yu's property A, then a band-dominated operator on X is Fredholm if and only if all of its limit operators are invertible. We also show that this always fails for metric spaces without property A.

math.FA↗

arXiv2014

Strong hyperbolicity

We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specifically, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at infinity, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(-1) spaces are strongly hyperbolic. On the way, we determine the best constant of hyperbolicity for the standard hyperbolic plane. We also show that the Green metric defined by a random walk on a hyperbolic group is strongly hyperbolic. A measure-theoretic consequence at the boundary is that the harmonic measure defined by a random walk is a visual Hausdorff measure.

math.GR↗

arXiv2013

On Rigidity of Roe algebras

Roe algebras are C*-algebras built using large-scale (or 'coarse') aspects of a metric space (X,d). In the special case that X=G is a finitely generated group and d is a word metric, the simplest Roe algebra associated to (G,d) is isomorphic to the reduced crossed product C*-algebra l^\infty(G)\rtimes G. Roe algebras are 'coarse invariants', in the sense that if X and Y are coarsely equivalent metric spaces, then their Roe algebras are isomorphic. Motivated in part by the coarse Baum-Connes conjecture, we ask if there is a converse to the above statement: that is, if X and Y are metric spaces with isomorphic Roe algebras, must X and Y be coarsely equivalent? We show that for very large classes of spaces the answer to this question, and some related questions, is 'yes'. This can be thought of as a 'C*-rigidity result': it shows that the Roe algebra construction preserves a large amount of information about the space, and is thus surprisingly 'rigid'. As an example of our results, in the group case we have that if G and H are finitely generated elementary amenable, hyperbolic, or linear, groups such that the crossed products l^\infty(G)\rtimes G and l^\infty(H)\rtimes H are isomorphic, then G and H are quasi-isometric.

math.OA↗

arXiv2012

Uniform Local Amenability

The main results of this paper show that various coarse (`large scale') geometric properties are closely related. In particular, we show that property A implies the operator norm localisation property, and thus that norms of operators associated to a very large class of metric spaces can be effectively estimated. The main tool is a new property called uniform local amenability. This property is easy to negate, which we use to study some `bad' spaces. We also generalise and reprove a theorem of Nowak relating amenability and asymptotic dimension in the quantitative setting.

math.MG↗

arXiv2011

Coarse non-amenability and coarse embeddings

We construct the first example of a coarsely non-amenable (= without Guoliang Yu's property A) metric space with bounded geometry which coarsely embeds into a Hilbert space.

math.GR↗

arXiv2011

Maximal and reduced Roe algebras of coarsely embeddable spaces

Gong, Wang and Yu introduced a maximal, or universal, version of the Roe C*-algebra associated to a metric space. We study the relationship between this maximal Roe algebra and the usual version, in both the uniform and non-uniform cases. The main result is that if a (uniformly discrete, bounded geometry) metric space X coarsely embeds in a Hilbert space, then the canonical map between the maximal and usual (uniform) Roe algebras induces an isomorphism on K-theory. We also give a simple proof that if X has property A, then the maximal and usual (uniform) Roe algebras are the same. These two results are natural coarse-geometric analogues of certain well-known implications of a-T-menability and amenability for group C*-algebras. The techniques used are E-theoretic, building on work of Higson-Kasparov-Trout and Yu.

math.KT↗

arXiv2010

Controlled coarse homology and isoperimetric inequalities

We study a coarse homology theory with prescribed growth conditions. For a finitely generated group G with the word length metric this homology theory turns out to be related to amenability of G. We characterize vanishing of a certain fundamental class in our homology in terms of an isoperimetric inequality on G and show that on any group at most linear control is needed for this class to vanish. The latter is a homological version of the classical Burnside problem for infinite groups, with a positive solution. As applications we characterize existence of primitives of the volume form with prescribed growth and show that coarse homology classes obstruct weighted Poincare inequalities.

math.GR↗

arXiv2009

Non-K-exact uniform Roe C*-algebras

We prove that uniform Roe C*-algebras associated to some expander graphs coming from discrete groups with property (τ) are not K-exact. In particular, we show that this is the case for the expander obtained as Cayley graphs of a sequence of alternating groups (with appropriately chosen generating sets).

math.OA↗

arXiv2009

Uniform version of Weyl-von Neumann theorem

We prove a "quantified" version of the Weyl-von Neumann theorem, more precisely, we estimate the ranks of approximants to compact operators appearing in the Voiculescu's theorem applied to commutative algebras. This allows considerable simplifications in uniform K-homology theory, namely it shows that one can represent all the uniform K-homology classes on a fixed Hilbert space with a fixed *-representation of C_0(X), for a large class of spaces X.

math.FA↗