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Piotr W. Nowak

Publications and source records attributed to Piotr W. Nowak.

At least 19 recordsLinked to original sources
arXiv2026

Property $(T)$ and nonlinearity of mapping class group quotients

We give a general method for proving Kazhdan's property $(T)$ for quotients $G/K_{[c+1]}$, where $G$ is countable, $K$ is a normal subgroup and $K_{[j]}$ denotes its lower central series. The method combines an affine realization of $G/[K,K]$, a contraction argument, and permanence for nilpotent normal subgroups. We apply it to the Torelli lower-central quotients $\mathrm{Mod}(Σ_g)/(\mathcal{T}_g)_{[c+1]}$ for $g\ge3$ and show that they have property $(T)$ for every $c\ge1$. We also prove property $(T)$ for $\mathrm{Aut}(F_3)/(\mathrm{IA}_3)_{[c+1]}$ for every $c\ge1$, relating these groups to the tame nilpotent images studied by Lubotzky and Pak. For the Torelli lower-central quotients with $g\ge3$ and $c\ge2$, every finite-dimensional complex representation has infinite kernel, and these quotients are not linear over any field.

math.GR↗
arXiv2026

Cocycles and positive functionals in higher cohomology

We establish and explore the correspondence between positive functionals and cocycles in higher unitary cohomology. We generalize the classical cocycle version of the Gelfand-Naimark-Segal construction to higher degrees and apply it to characterize vanishing of higher unitary cohomology as an extension property for positive functionals. We also prove that under mild conditions the algebraic spectral gap for the one sided Laplacian characterizes cohomological vanishing instead of reducedness of unitary cohomology

math.AT↗
arXiv2023

On order units in the augmentation ideal

We study order units in the real group ring and the augmentation ideal, as well as in matrix algebras. We identify an infinite family of order units in the powers of the augmentation ideal, that includes the Laplacian, and show that these order units are naturally obtained via cohomological operations from more simpler diagonal order units in matrix algebras.

math.GR↗
arXiv2023

Coboundary expansion and Gromov hyperbolicity

We prove that if a compact $n$-manifold admits a sequence of residual covers that form a coboundary expander in dimension $n-2$, then the manifold has Gromov-hyperbolic fundamental group. In particular, residual sequences of covers of non-hyperbolic compact connected irreducible 3-manifolds are not 1-coboundary expanders.

math.GT↗
arXiv2023

On (bi)reversible automata generating lamplighter groups

For any nontrivial abelian group $\mathbb{X}$ we construct a reversible (bireversible in case the order of $\mathbb{X}$ is odd) automaton such that its set of states and alphabet are identified with $\mathbb{X}$, transition and output functions are defined via the left and the right regular actions correspondingly and its group splits into the restricted wreath product $\mathbb{X} \wr \mathbb{Z}$, i.e. is a lamplighter group.

math.GR↗
arXiv2021

On property (T) for $\operatorname{Aut}(F_n)$ and $\operatorname{SL}_n(\mathbb{Z})$

We prove that $\operatorname{Aut}(F_n)$ has Kazhdan's property (T) for every $n \geqslant 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n\geqslant 5$. We also provide explicit lower bounds for the Kazhdan constants of $\operatorname{SAut}(F_n)$ (with $n \geqslant 6$) and of $\operatorname{SL}_n(\mathbb{Z})$ (with $n \geqslant 3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n > 6$.

math.OA↗
arXiv2020

Higher Kazhdan projections, $\ell_2$-Betti numbers and Baum-Connes conjectures

We introduce higher-dimensional analogs of Kazhdan projections in matrix algebras over group $C^*$-algebras and Roe algebras. These projections are constructed in the framework of cohomology with coefficients in unitary representations and in certain cases give rise to non-trivial $K$-theory classes. We apply the higher Kazhdan projections to establish a relation between $\ell_2$-Betti numbers of a group and surjectivity of different Baum-Connes type assembly maps.

math.OA↗
arXiv2018

Singular subgroups in $\tilde{A}_2$-groups and their von Neumann algebras

We show that certain amenable subgroups inside $\tilde{A}_2$-groups are singular in the sense of Boutonnet and Carderi. This gives a new family of examples of singular group von Neumann subalgebras. We also give a geometric proof that if $G$ is an acylindrically hyperbolic group, $H$ is an infinite amenable subgroup containing a loxodromic element, then $H<G$ is singular. Finally, we present (counter)examples to show both situations happen concerning maximal amenability of $LH$ inside $LG$ if $H$ does not contain loxodromic elements.

math.OA↗
arXiv2017

Kazhdan projections, random walks and ergodic theorems

In this paper we investigate generalizations of Kazhdan's property $(T)$ to the setting of uniformly convex Banach spaces. We explain the interplay between the existence of spectral gaps and that of Kazhdan projections. Our methods employ Markov operators associated to a random walk on the group, for which we provide new norm estimates and convergence results. They exhibit useful properties and flexibility, and allow to view Kazhdan projections in Banach spaces as natural objects associated to random walks on groups. We give a number of applications of these results. In particular, we address several open questions. We give a direct comparison of properties $(TE)$ and $FE$ with Lafforgue's reinforced Banach property $(T)$; we obtain shrinking target theorems for orbits of Kazhdan groups; finally, answering a question of Willett and Yu we construct non-compact ghost projections for warped cones. In this last case we conjecture that such warped cones provide counterexamples to the coarse Baum-Connes conjecture.

math.GR↗
arXiv2016

Group 1-cohomology is complemented

We show a structural property of cohomology with coefficients in an isometric representation on a uniformly convex Banach space: if the cohomology group $H^1(G,π)$ is reduced, then, up to an isomorphism, it is a closed complemented, subspace of the space of cocycles and its complement is the subspace of coboundaries.

math.GR↗
arXiv2016

Warped cones and spectral gaps

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.

math.MG↗
arXiv2016

Eilenberg swindles and higher large scale homology of products of trees

We show that uniformly finite homology of products of $n$ trees vanishes in all degrees except degree $n$, where it is infinite dimensional. Our method is geometric and applies to several large scale homology theories, including almost equivariant homology and controlled coarse homology. As an application we determine group homology with $\ell_{\infty}$-coefficients of lattices in products of trees. We also show a characterization of amenability in terms of 1-homology and construct aperiodic tilings using higher homology.

math.GT↗