arXiv subjects
Publications and source records attributed to Andreas Thom.
We prove new upper bounds for the length of laws that hold for all groups of size at most $n$ -- improving on previous results of Bou-Rabee and Kassabov-Matucci. The methods make use of the classification of finite simple groups. Stronger bounds are proved in case the groups are assumed to be nilpotent or solvable.
We study the question how quickly products of a fixed conjugacy class in the projective unitary group of a II${}_1$-factor von Neumann algebra cover the entire group. Our result is that the number of factors that are needed is essentially as small as permitted by the $1$-norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups. As an application of the techniques, we prove that every homomorphism from the projective unitary group of a II${}_1$-factor to a polish SIN group is continuous. Moreover, we show that the projective unitary group of a II${}_1$-factor carries a unique polish group topology.
We establish a characterization of amenability for general Hausdorff topological groups in terms of matchings with respect to finite uniform coverings. Furthermore, we prove that it suffices to just consider two-element uniform coverings. We also show that extremely amenable as well as compactly approximable topological groups satisfy a perfect matching property condition -- the latter even with regard to arbitrary uniform coverings. Finally, we prove that the automorphism group of a Fraïssé limit of a metric Fraïssé class is amenable if and only if the considered metric Fraïssé class has a certain Ramsey-type matching property.
We provide an alternative proof for the extreme amenability of the unitary group of the hyperfinite II${}_1$-factor von Neumann algebra, endowed with the strong operator topology.
A palindrome is a word that reads the same left-to-right as right-to-left. We show that every simple group has a finite generating set $X$, such that every element of it can be written as a palindrome in the letters of $X$. Moreover, every simple group has palindromic width $pw(G,X)=1$, where $X$ only differs by at most one Nielsen-transformation from any given generating set. On the contrary, we prove that all non-abelian finite simple groups $G$ also have a generating set $S$ with $pw(G,S)>1$. As a by-product of our work we also obtain that every just-infinite group has finite palindromic width with respect to a finite generating set. This provides first examples of groups with finite palindromic width but infinite commutator width.
Following an idea of Ozawa, we give a new proof of Kazhdan's property (T) for ${\rm SL}(3,\mathbb Z)$, by showing that $Δ^2- \frac{1}{6} Δ$ is a hermitian sum of squares in the group algebra, where $Δ$ is the unnormalized Laplace operator with respect to the natural generating set. This corresponds to a spectral gap of $\frac{1}{72}\sim 0.014$ for the associated random walk operator. The sum of squares representation was found numerically by a semidefinite programming algorithm, and then turned into an exact symbolic representation, provided in an attached Mathematica file.
We provide new bounds for the divisibility function of the free group F_2 and construct short laws for the symmetric groups Sym(n). The construction is random and relies on the classification of the finite simple groups. We also give bounds on the length of laws for finite simple groups of Lie type.
To any positive contraction $Q$ on $\ell^2(W)$, there is associated a determinantal probability measure ${\mathbf P}^Q$ on $2^W$, where $W$ is a denumerable set. Let $Γ$ be a countable sofic finitely generated group and $G = (Γ, \mathsf{E})$ be a Cayley graph of $Γ$. We show that if $Q_1$ and $Q_2$ are two $Γ$-equivariant positive contractions on $\ell^2(Γ)$ or on $\ell^2(\mathsf{E})$ with $Q_1 \le Q_2$, then there exists a $Γ$-invariant monotone coupling of the corresponding determinantal probability measures witnessing the stochastic domination ${\bf P}^{Q_1} \preccurlyeq {\bf P}^{Q_2}$. In particular, this applies to the wired and free uniform spanning forests, which was known before only when $Γ$ is residually amenable. In the case of spanning forests, we also give a second more explicit proof, which has the advantage of showing an explicit way to create the free uniform spanning forest as a limit over a sofic approximation. Another consequence of our main result is to prove that all determinantal probability measures ${\bf P}^Q$ as above are ${\bar d}$-limits of finitely dependent processes. Thus, when $Γ$ is amenable, ${\bf P}^Q$ is isomorphic to a Bernoulli shift, which was known before only when $Γ$ is abelian. We also prove analogous results for sofic unimodular random rooted graphs.
In this paper we study characters on special linear groups SL_n(R), where R is either an infinite field or the localization of an order in a number field. We give several applications to the theory of measure preserving actions, operator-algebraic superrigidity, and almost homomorphisms.
Some new results on metric ultraproducts of finite simple groups are presented. Suppose that G is such a group, defined in terms of a non-principal ultrafilter ω on N and a sequence {(G_i)_{i \in N}} of finite simple groups, and that G is neither finite nor a Chevalley group over an infinite field. Then G is isomorphic to an ultraproduct of alternating groups or to an ultraproduct of finite simple classical groups. The isomorphism type of G determines which of these two cases arises, and, in the latter case, the ω-limit of the characteristics of the groups Gi. Moreover G is a complete path-connected group with respect to the natural metric on G.
In this note we study sets of normal generators of finitely presented residually $p$-finite groups. We show that if an infinite, finitely presented, residually $p$-finite group $G$ is normally generated by $g_1,\dots,g_k$ with order $n_1,\dots,n_k \in \{1,2,\dots \} \cup \{\infty \}$, then $$β_1^{(2)}(G) \leq k-1-\sum_{i=1}^{k} \frac1{n_i},$$ where $β_1^{(2)}(G)$ denotes the first $\ell^2$-Betti number of $G$. We also show that any $k$-generated group with $β_1^{(2)}(G) \geq k-1-\varepsilon$ must have girth greater than or equal $1/\varepsilon$.
We provide upper and lower bounds on the length of the shortest non-trivial element in the derived series and lower central series in the free group on two generators. The techniques are used to provide new estimates on the nilpotent residual finiteness growth and on almost laws for compact groups.
Studying inequalities between subgraph- or homomorphism-densities is an important topic in graph theory. Sums of squares techniques have proven useful in dealing with such questions. Using an approach from real algebraic geometry, we strengthen a Positivstellensatz for simple quantum graphs by Lovász and Szegedy, and we prove several new Positivstellensätze for nonnegativity of quantum multigraphs. We provide new examples and counterexamples.
We show that for any amenable group Γand any ZΓ-module M of type FL with vanishing Euler characteristic, the entropy of the natural Γ-action on the Pontryagin dual of M is equal to the L2-torsion of M. As a particular case, the entropy of the principal algebraic action associated with the module ZΓ/ZΓf is equal to the logarithm of the Fuglede-Kadison determinant of f whenever f is a non-zero-divisor in ZΓ. This confirms a conjecture of Deninger. As a key step in the proof we provide a general Szegő-type approximation theorem for the Fuglede-Kadison determinant on the group von Neumann algebra of an amenable group. As a consequence of the equality between L2-torsion and entropy, we show that the L2-torsion of a non-trivial amenable group with finite classifying space vanishes. This was conjectured by Lück. Finally, we establish a Milnor-Turaev formula for the L2-torsion of a finite Δ-acyclic chain complex.
We show that for any finitely generated non-amenable group and any $\varepsilon>0$, there exists some finite symmetric generating set with spectral radius less than $\varepsilon$. We give applications to percolation theory and the theory of operator spaces.
We give a lower bound on the expected degree of the free minimal spanning forest of a vertex transitive graph in terms of its spectral radius. This result answers a question of Lyons-Peres-Schramm and simplifies the Gaboriau-Lyons proof of the measurable-group-theoretic solution to von Neumann's problem. In the second part we study a relative version of the free minimal spanning forest. As a consequence of this study we can show that non-torsion unitarizable groups have fixed price one.
We prove that the lattice of normal subgroups of ultraproducts of compact simple non-abelian groups is distributive. In the case of ultraproducts of finite simple groups or compact connected simple Lie groups of bounded rank the set of normal subgroups is shown to be linearly ordered by inclusion.
In this short note we study the entropy for algebraic actions of certain amenable groups. The possible values for this entropy are studied. Various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant. We provide a natural decomposition of the entropy into summands contributed by individual primes and a summand corresponding to infinity. These results extend previous work by Lind and Ward on p-adic entropy.