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Maciej Malicki

Publications and source records attributed to Maciej Malicki.

29 records · Page 2Linked to original sources
arXiv2017

Property (T), finite-dimensional representations, and generic representations

Let $G$ be a discrete group with property (T). It is a standard fact that, in a unitary representation of $G$ on a Hilbert space $\mathcal{H}$, almost invariant vectors are close to invariant vectors, in a quantitative way. We begin by showing that, if a unitary representation has some vector whose coefficient function is close to a coefficient function of some finite-dimensional unitary representation $σ$, then the vector is close to a sub-representation isomorphic to $σ$: this makes quantitative a result of P.S. Wang [Wa]. We use that to give a new proof of a result by D. Kerr, H. Li and M. Pichot [KLP], that a group $G$ with property (T) and such that $C^*(G)$ is residually finite-dimensional, admits a unitary representation which is generic (i.e. the orbit of this representation in $\mathrm{Rep}(G,\mathcal{H})$ under the unitary group $U(\mathcal{H})$ is comeager). We also show that, under the same assumptions, the set of representations equivalent to a Koopman representation is comeager in $\mathrm{Rep}(G,\mathcal{H})$.

math.GR↗
arXiv2016

Matheme and mathematics. On the main concepts of the philosophy of Alain Badiou

In this paper, I present a critical discussion of mathematical arguments employed in the philosophy of event of Alain Badiou. On the basis of "Being and Event" as well as his other writings, I analyze the main notions of his philosophy such as the indiscernible, the undecidable, and the unnameable. The focus of my analysis is both on their mathematical consistency, and their philosophical consequences. I argue that the mathematical approach developed by Badiou is seriously defective, and, as a result, that it cannot serve as an ontological basis for the concept of event as presented in "Being and Event".

math.LO↗
arXiv2016

Abelian pro-countable groups and non-Borel orbit equivalence relations

We study topological groups that can be defined as Polish, pro-countable abelian groups, as non-archimedean abelian groups or as quasi-countable abelian groups, i.e., Polish subdirect products of countable, discrete groups, endowed with the product topology. We characterize tame groups in this class, i.e., groups such that all orbit equivalence relations induced by their continuous actions on Polish spaces are Borel, and relatively tame groups $G$, i.e., groups such that every diagonal action $α\times β$ of $G$ induces a Borel orbit equivalence relation, provided that the actions $α$, $β$ of $G$ are continuous, and induce Borel orbit equivalence relations.

math.LO↗
arXiv2015

Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures

We define a simple criterion for a homogeneous, complete metric structure $X$ that implies that the automorphism group $\mbox{Aut}(X)$ satisfies all the main consequences of the existence of ample generics: it has the small index property, the automatic continuity property, and uncountable cofinality for non-open subgroups. Then we verify it for the Urysohn space $\mbox{U}$, the Lebesgue probability measure algebra $\mbox{MALG}$, and the Hilbert space $\ell_2$, thus proving that $\mbox{Iso}(\mbox{U})$, $\mbox{Aut}(\mbox{MALG})$, $U(\ell_2)$, and $O(\ell_2)$ share these properties. We also formulate a condition for $X$ which implies that every homomorphism of $\mbox{Aut}(X)$ into a separable group $K$ with a left-invariant, complete metric, is trivial, and we verify it for $\mbox{U}$, and $\ell_2$.

math.LO↗
arXiv2015

Generic elements in isometry groups of Polish ultrametric spaces

This paper presents a study of generic elements in full isometry groups of Polish ultrametric spaces. We obtain a complete characterization of Polish ultrametric spaces X whose isometry group Iso(X) contains an open subgroup H with ample generics. It also gives a characterization of the existence of an open subgroup in Iso(X) with a comeager conjugacy class. We also study the transfinite sequence defined by the projection of a Polish ultrametric space X on the ultrametric space of orbits of X under the action of Iso(X).

math.GR↗
arXiv2015

Abelian pro-countable groups and orbit equivalence relations

We study groups that can be defined as Polish, pro-countable groups, as non-archimedean groups with an invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable, discrete groups, endowed with the product topology. We show, among other results, that for every non-locally compact, abelian quasi-countable group G there exists a closed subgroup L of G, and a closed, non-locally compact subgroup K of G/L which is a direct product of discrete, countable groups. As an application we prove that for every abelian Polish group G of the form H/L, where H,L are closed subgroups of Iso(X) and X is a locally compact separable metric space (e.g., G is abelian, quasi-countable), G is locally compact iff every continuous action of G on a Polish space Y induces an orbit equivalence relation that is reducible to an equivalence relation with countable classes.

math.GR↗
arXiv2015

An example of a non non-archimedean Polish group with ample generics

For an analytic $P$-ideal $I$, $S_I$ is the Polish group of all the permutations of $\mathbb{N}$ whose support is in $I$, with Polish topology given by the corresponding submeasure on $I$. We show that if $\mbox{Fin} \subsetneq I$, then $S_I$ has ample generics. This implies that there exists a non non-archimedean Polish group with ample generics.

math.GR↗
arXiv2013

Cooperative Boolean systems with generically long attractors II

We prove that cooperativity in Boolean networks precludes a strong notion of sensitive dependence on initial conditions. Weaker notions of sensitive dependence are shown to be consistent with cooperativity, but if each regulatory functions is binary AND or binary OR, in N-dimensional networks they impose an upper bound of approximately sqrt(3)^N on the lengths of attractors that can be reached from a fraction p approaching 1 of initial conditions. The upper bound is shown to be sharp.

math.DS↗
arXiv2013

Cooperative Boolean systems with generically long attractors I

We study the class of cooperative Boolean networks whose only regulatory functions are COPY, binary AND, and binary OR. We prove that for all sufficiently large N and c < 2 there exist Boolean networks in this class that have an attractor of length > c^N whose basin of attraction comprises an arbitrarily large fraction of the state space. The existence of such networks contrasts with results on various other types of dynamical systems that show nongenericity or absence of non-steady state attractors under the assumption of cooperativity.

math.DS↗
arXiv2011

Trees, unsplittability, property (FA) and the likes

The paper is devoted to a study of certain fixed point properties, and their relatives, in the context of full automorphism groups of countable rooted trees. Namely, we study Serre's property (FA'), also called unsplittability, property (FA), the uncountable strong cofinality, and ample generics. We give a new proof of a theorem of Psaltis to the extent that automorphism groups of rooted trees are unsplittable, and show under what circumstances automorphism groups of rooted trees have uncountable strong cofinality (and thus property (FA).) Also, we prove a necessary and sufficient condition for automorphism groups of rooted trees to have ample generics. This very strong property has interesting implications, such as the small index property, and continuity of homomorphisms. As an application, we analyze the relationship between two generalizations (discovered by Psaltis and Forester) of an interesting rigidity result on locally finite trees, originally proved by Bass and Lubotzky.

math.GR↗