Related papers: Is being a higher rank lattice a first order prope…

For an integer $M\geq 2$ and a finite group $G$, an element $\alpha\in G$ is called an $M$-th power if it satisfies $A^M=\alpha$ for some $A\in G$. In this article, we will deal with the case when $G$ is finite symplectic or orthogonal…

Let $G$ be a right-angled Artin group with defining graph $\Gamma$ and let $H$ be a finitely generated group quasi-isometric to $G(\Gamma)$. We show if $G$ satisfies (1) its outer automorphism group is finite; (2) $\Gamma$ does not have…

We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first…

A group $\Gamma$ is said to be periodic if for any $g$ in $\Gamma$ there is a positive integer $n$ with $g^n=id$. We first prove that a finitely generated periodic group acting on the 2-sphere $\SS^2$ by $C^1$-diffeomorphisms with a finite…

We prove that if $\Gamma$ is a lattice in the group of isometries of a symmetric space of non-compact type without euclidean factors, then the virtual cohomological dimension of $\Gamma$ equals its proper geometric dimension.

Let us say that a Cayley graph $\Gamma$ of a group $G$ of order $n$ is a Cerny Cayley graph if every synchronizing automaton containing $\Gamma$ as a subgraph with the same vertex set admits a synchronizing word of length at most $(n-1)^2$.…

For a finitely generated lawless group $\Gamma$ and $n \in \mathbb{N}$, let $\mathcal{A}_{\Gamma} (n)$ be the minimal positive integer $M_n$ such that for all nontrivial reduced words $w$ of length at most $n$ in the free group of fixed…

A profinite group is called small if it has only finitely many open subgroups of index n for each positive integer n. We show that every Frattini cover of a small profinite group is small. A profinite group is called strongly complete if…

We consider largeness of groups given by a presentation of deficiency 1, where the group is respectively free-by-cyclic, LERF or 1-relator. We give the first examples of (finitely generated free)-by-(infinite cyclic) word hyperbolic groups…

An even unimodular 72-dimensional lattice $\Gamma $ having minimum 8 is constructed as a tensor product of the Barnes lattice and the Leech lattice over the ring of integers in the imaginary quadratic number field with discriminant $-7$.…

A free-by-cyclic group $F_N\rtimes_\phi\mathbb{Z}$ has non-trivial centre if and only if $[\phi]$ has finite order in ${\rm{Out}}(F_N)$. We establish a profinite ridigity result for such groups: if $\Gamma_1$ is a free-by-cyclic group with…

It is well-known that a finitely generated group $\Gamma$ has Kazhdan's property (T) if and only if the Laplacian element $\Delta$ in ${\mathbb R}[\Gamma]$ has a spectral gap. In this paper, we prove that this phenomenon is witnessed in…

Let X(F,G) be the G-character variety of F where G is a rank 1 complex affine algebraic group and F is a finitely presentable discrete group. We describe an algorithm, which we implement in Mathematica, SageMath, and in Python, that takes a…

We consider the problems of deciding whether an input graph can be modified by removing/adding at most k vertices/edges such that the result of the modification satisfies some property definable in first-order logic. We establish a number…

The spectrum $\omega(G)$ of a finite group $G$ is the set of element orders of $G$. Finite groups $G$ and $H$ are isospectral if their spectra coincide. Suppose that $L$ is a simple classical group of sufficiently large dimension (the lower…

In a finite group $ G $, $ \psi(G) $ denotes the sum of element orders of $ G $. A finite group $ G $ is said to be a $\mathscr{B}_{\psi}$-group if $ \psi(H) < |G| $ for any proper subgroup $ H $ of $ G $. In \cite{Lazorec} Lazorec asked:…

Let $G$ be a finite group, $n$ a positive integer. $\pi(n)$ denotes the set of all prime divisors of $n$ and $\pi(G)=\pi(|G|)$. The prime graph $\Gamma(G)$ of $G$, defined by Grenberg and Kegel, is a graph whose vertex set is $\pi(G)$, two…

In this sequel paper, we continue the analysis of the prime order element graph $\Gamma(G)$ of a finite group $G$, where vertices are elements of $G$ and edges connect distinct elements $x, y$ satisfying $\circ(xy) = p$ for some prime $p$.…

We lay the foundations of the first-order model theory of Coxeter groups. Firstly, with the exception of the $2$-spherical non-affine case (which we leave open), we characterize the superstable Coxeter groups of finite rank, which we show…

Let $(W,S)$ be an arbitrary Coxeter system. For each word $\omega$ in the generators we define a partial order--called the {\sf $\omega$-sorting order}--on the set of group elements $W_\omega\subseteq W$ that occur as subwords of $\omega$.…