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Andreas Thom

Publications and source records attributed to Andreas Thom.

At least 73 records · Page 4Linked to original sources
arXiv2017

A note on the normal subgroup lattice of ultraproducts of finite quasisimple groups

In [A. Stolz and A. Thom, On the lattice of normal subgroups in ultraproducts of compact simple groups, PLMS 108(1), 2014] it was stated that the lattice of normal subgroups of an ultraproduct of finite simple groups is always linearly ordered. This is false in this form in most cases for classical groups of Lie type. We correct the statement and point out a version of 'relative' bounded generation results for classical quasisimple groups and its implications on the structure of the lattice of normal subgroups of an ultraproduct of quasisimple groups.

math.GR↗
arXiv2017

Unitarizability, Maurey--Nikishin factorization, and Polish groups of finite type

Let $Γ$ be a countable discrete group, and let $π\colon Γ\to {\rm{GL}}(H)$ be a representation of $Γ$ by invertible operators on a separable Hilbert space $H$. We show that the semidirect product group $G=H\rtimes_πΓ$ is SIN ($G$ admits a two-sided invariant metric compatible with its topology) and unitarily representable ($G$ embeds into the unitary group $\mathcal{U}(\ell^2(\mathbb N))$), if and only if $π$ is uniformly bounded, and that $π$ is unitarizable if and only if $G$ is of finite type: that is, $G$ embeds into the unitary group of a II$_1$-factor. Consequently, we show that a unitarily representable Polish SIN groups need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey--Nikishin factorization theorem for continuous maps from a Hilbert space to the space $L^0(X,m)$ of all measurable maps on a probability space.

math.OA↗
arXiv2017

Operator algebraic approach to inverse and stability theorems for amenable groups

We prove an inverse theorem for the Gowers $U^2$-norm for maps $G\to\mathcal M$ from an countable, discrete, amenable group $G$ into a von Neumann algebra $\mathcal M$ equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm $\Vert\cdot\Vert$. We use this result to prove a stability result for unitary-valued $\varepsilon$-representations $G\to\mathcal U(\mathcal M)$ with respect to $\Vert\cdot \Vert$.

math.OA↗
arXiv2017

Short Laws for Finite Groups and Residual Finiteness Growth

We prove that for every $n \in \mathbb{N}$ and $δ>0$ there exists a word $w_n \in F_2$ of length $n^{2/3} \log(n)^{3+δ}$ which is a law for every finite group of order at most $n$. This improves upon the main result of [A. Thom, About the length of laws for finite groups, Isr. J. Math.]. As an application we prove a new lower bound on the residual finiteness growth of non-abelian free groups.

math.GR↗
arXiv2017

An exotic group as limit of finite special linear groups

We consider the Polish group obtained as the rank-completion of an inductive limit of finite special linear groups. This Polish group is topologically simple modulo its center, it is extremely amenable and has no non-trivial strongly continuous unitary representation on a Hilbert space.

math.GR↗
arXiv2017

Some remarks on finitarily approximable groups

The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite solvable groups with arbitrary invariant length function. We answer this question by showing that any non-trivial finitely generated perfect group does not have this property, generalizing a counterexample of Howie. Moreover, we discuss the question which connected Lie groups can be embedded into a metric ultraproduct of finite groups with invariant length function. We prove that these are precisely the abelian ones, providing a negative answer to a question of Doucha. Referring to a problem of Zilber, we show that a the identity component of a Lie group, whose topology is generated by an invariant length function and which is an abstract quotient of a product of finite groups, has to be abelian. Both of these last two facts give an alternative proof of a result of Turing. Finally, we solve a conjecture of Pillay by proving that the identity component of a compactification of a pseudofinite group must be abelian as well. All results of this article are applications of theorems on generators and commutators in finite groups by the first author and Segal.

math.GR↗
arXiv2017

Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization

Containment problems for polytopes and spectrahedra appear in various applications, such as linear and semidefinite programming, combinatorics, convexity and stability analysis of differential equations. This paper explores the theoretical background of a method proposed by Ben-Tal and Nemirovksi. Their method provides a strengthening of the containment problem, that is algorithmically well tractable. To analyze this method, we study abstract operator systems, and investigate when they have a finite-dimensional concrete realization. Our results give some profound insight into their approach. They imply that when testing the inclusion of a fixed polyhedral cone in an arbitrary spectrahedron, the strengthening is tight if and only if the polyhedral cone is a simplex. This is true independent of the representation of the polytope. We also deduce error bounds in the other cases, simplifying and extending recent results by various authors.

math.FA↗
arXiv2017

Cantor systems and quasi-isometry of groups

The purpose of this note is twofold. In the first part we observe that two finitely generated non-amenable groups are quasi-isometric if and only if they admit topologically orbit equivalent Cantor minimal actions. In particular, free groups if different rank admit topologically orbit equivalent Cantor minimal actions unlike in the measurable setting. In the second part we introduce the measured orbit equivalence category of a Cantor minimal system and construct (in certain cases) a representation of this category on the category of finite-dimensional vector spaces. This gives rise to novel fundamental invariants of the orbit equivalence relation together with an ergodic invariant probability measure.

math.DS↗
arXiv2016

On Følner sets in topological groups

We extend Følner's amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left translation action can be approximated in a uniform manner by amenable actions on the set $G$. As applications we obtain a topological version of Whyte's geometric solution to the von Neumann problem and provide an affirmative answer to a question posed by Rosendal.

math.GR↗
arXiv2016

About the solvability of matrix polynomial equations

We study self-adjoint matrix polynomial equations in a single variable and prove existence of self-adjoint solutions under some assumptions on the leading form. Our main result is that any self-adjoint matrix polynomial equation of odd degree with non-degenerate leading form can be solved in self-adjoint matrices. We also study equations of even degree and equations in many variables.

math.RA↗
arXiv2016

Some geometric properties of metric ultraproducts of finite simple groups

In this article we prove some previously announced results about metric ultraproducts of finite simple groups. We show that any non-discrete metric ultraproduct of alternating or special linear groups is a geodesic metric space. For more general non-discrete metric ultraproducts of finite simple groups, we are able to establish path-connectedness. As expected, these global properties reflect asymptotic properties of various families of finite simple groups.

math.GR↗
arXiv2016

Bounded normal generation for projective unitary groups of certain infinite operator algebras

We study the question how quickly products of a fixed conjugacy class cover the entire group in the projective unitary group of the connected component of the identity of the Calkin algebra, as well as the projective unitary group of a factor von Neumann algebra of type III. Our result is that the number of factors that are needed is as small as permitted by the (essential) operator norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups and analogous results for unitary groups of II_1-factors.

math.FA↗
arXiv2016

New topological methods to solve equations over groups

We show that the equation associated with a group word $w \in G \ast {\mathbf F}_2$ can be solved over a hyperlinear group $G$ if its content - that is its augmentation in ${\mathbf F}_2$ - does not lie in the second term of the lower central series of ${\mathbf F}_2$. Moreover, if $G$ is finite, then a solution can be found in a finite extension of $G$. The method of proof extends techniques developed by Gerstenhaber and Rothaus, and uses computations in $p$-local homotopy theory and cohomology of compact Lie groups.

math.GR↗
arXiv2016

Quadratic Modules, C*-Algebras, and Free Convexity

Given a quadratic module, we construct its universal C*-algebra, and then use methods and notions from the theory of C*-algebras to study the quadratic module. We define residually finite-dimensional quadratic modules, and characterize them in various ways, in particular via a Positivstellensatz. We give unified proofs for several existing strong Positivstellensätze, and prove some new ones. Our approach also leads naturally to interesting new examples in free convexity. We show that the usual notion of a free convex hull is not able to detect residual finite-dimensionality. We thus propose a new notion of free convexity, which is coordinate-free. We characterize semialgebraicity of free convex hulls of semialgebraic sets, and show that they are not always semialgebraic, even at scalar level. This also shows that the membership problem for quadratic modules has a negative answer in the non-commutative setup.

math.OA↗
arXiv2016

Jost Bürgi's Method for Calculating Sines

For a long time it has been known that in the 16th century the Swiss mathematician Jost Bürgi found a new method for calculating sines, but no information about the details has been available. Recently a manuscript written by Bürgi himself has come to light in which he explains his algorithm. It is totally different from the traditional procedure which was used until the 17th century. In the first part of the article the standard method is explained which was rooted in Greek antiquity with Ptolemy's computation of chords and which was used in the Arabic-Islamic tradition and in the Western European Middle Ages for calculating chords as well as sines. The main part of the article deals with Bürgi's way. By only using additions and halving, his procedure is elementary and it converges quickly. Bürgi does not explain why his method is correct, but in the last part of the article a modern proof for the correctness of his algorithm is provided.

math.HO↗
arXiv2015

Equivariant characteristic forms in the Cartan model and Borel equivariant cohomology

We show the compatibility of the differential geometric and the topological construction of equivariant characteristic classes for compact Lie groups. Our analysis motivates a differential geometric construction for equivariant characteristic classes in the non-compact case. This compatibility is generally assumed and used in various cases, but there is only a proof for compact connected Lie groups in the literature, see the work of Raoul Bott and Loring Tu. Our proof applies and generalizes ideas of Johan Dupont and Ezra Getzler.

math.AT↗
arXiv2015

Equivariant Differential Cohomology

The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de Rham cocycles -- known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: The definition of equivariant characteristic forms in the Cartan model due to Nicole Berline and Michèle Vergne motivates a refinement of equivariant integral cohomology by all Cartan cocycles. In view of this, we will also review previous definitions critically, in particular the one given by Kiyonori Gomi.

math.DG↗