Related papers: A continuum of non-measure equivalent groups

For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable…

A unital $\ell$-group $(G,u)$ is an abelian group $G$ equipped with a translation-invariant lattice-order and a distinguished element $u$, called order-unit, whose positive integer multiples eventually dominate each element of $G$. We…

We study generic properties of topological groups in the sense of Baire category. First we investigate countably infinite (discrete) groups. We extend a classical result of B. H. Neumann, H. Simmons and A. Macintyre on algebraically closed…

Let $d \geq 2$ be an integer. We conjecture that there is a finitely generated perfect group whose homomorphic images include all finite $d$-generated perfect groups. We prove a special case of this conjecture for the finite perfect groups…

Using the Generalized Maximium Entropy Principle based on the nonextensive q entropy a new family of random matrix ensembles is generated. This family unifies previous extensions of Random Matrix Theory and gives rise to an orthogonal…

In relation to the Erd\H os similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are…

We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x in G and a coset of G modulo its socle, the probability that x and a random element of the coset generate a subgroup…

This paper aims to investigate the self-similarity property in finitely-generated torsion-free nilpotent groups. We establish connections between geometric equivalence and self-similarity in these groups. Moreover, we show that any…

We present a construction of complete self-dual Einstein metrics of negative scalar curvature on an uncountable family of manifolds of infinite topological type, which are enumerated by continued fraction expansions of irrational numbers.…

Let $G$ be an additive abelian group. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ of terms from $G$ is a plus-minus weighted zero-sum sequence if there are $\varepsilon_1, \ldots, \varepsilon_{\ell} \in \{-1, 1\}$ such that…

We classify finite groups $G$, such that the group algebra, $\mathbb{Q}G$ (over the field of rational numbers $\mathbb{Q}$), is the direct product of the group algebra $\mathbb{Q}[G/N]$ of a proper factor group $G/N$, and some division…

If $G$ is a nilpotent group with a balanced presentation and $G\not\cong\mathbb{Z}^3$ then $\beta_1(G;\mathbb{Q})\leq2$ \cite{Hi22}. We show that if such a group $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{Z}^2$ then…

Motivated by a question of Bumagin and Wise, we construct a continuum of finitely generated, residually finite groups whose outer automorphism groups are pairwise non-isomorphic finitely generated, non-recursively-presentable groups. These…

It is shown that there exist finitely generated infinite simple groups of infinite commutator width and infinite square width on which there exists no stably unbounded conjugation-invariant norm, and in particular stable commutator length…

The problem of the existence of non-pseudo-$\aleph_1$-compact $\mathbb R$-factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than $\omega_1$. Closely related results concerning the…

In 2022, using methods from ergodic theory, Kra, Moreira, Richter, and Robertson resolved a longstanding conjecture of Erd\H{o}s about sumsets in large subsets of the natural numbers. In this paper, we extend this result to several…

We consider ergodic $\mathrm{Sym}(\mathbb{N})$-invariant probability measures on the space of $L$-structures with domain $\mathbb{N}$ (for $L$ a countable relational language), and call such a measure a properly ergodic structure when no…

We provide some characterizations of precompact abelian groups $G$ whose dual group $G_p^\wedge$ endowed with the pointwise convergence topology on elements of $G$ contains a nontrivial convergent sequence. In the special case of precompact…

We consider the oriented graph whose vertices are isomorphism classes of finitely generated groups, with an edge from G to H if, for some generating set T in H and some sequence of generating sets S_i in G, the marked balls of radius i in…

Let F be a separable uniformly bounded family of measurable functions on a standard measurable space, and let N_{[]}(F,\epsilon,\mu) be the smallest number of \epsilon-brackets in L^1(\mu) needed to cover F. The following are equivalent: 1.…