Related papers: An incoherent simple group

To each finitely generated group $G$, we associate a quasi-isometric invariant called the \emph{Dehn spectrum} of $G$. If $G$ is finitely presented, our invariant is closely related to the Dehn function of $G$, but provides more information…

A finitely generated group or monoid is said to be context-free if it has context-free word problem. In this note, we give an example of a context-free monoid, none of whose maximal subgroups are finitely generated. This answers a question…

The generalised Fitting subgroup of a finite group is the group generated by all subnormal subgroups that are either nilpotent or quasisimple. The importance of this subgroup in finite group theory stems from the fact that it always…

A finitely generated group $G$ is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety $\mathcal{UV}$, where $\mathcal{U}$ (respectively,…

A countable group is residually finite if every nontrivial element can act nontrivially on a finite set. When a group fails to be residually finite, we might want to measure how drastically it fails - it could be that only finitely many…

This note serves as a short and reader-friendly introduction to twisted Brin-Thompson groups, which were recently constructed by Belk and the author to provide a family of simple groups with a variety of interesting properties. Most…

We define for discrete finitely presented groups a new property related to their asymptotic representations. Namely we say that a groups has the property AGA if every almost representation generates an asymptotic representation. We give…

We prove that any non-cocompact irreducible lattice in a higher rank semi-simple Lie group contains a subgroup of finite index, which has three generators.

We say that a subgroup $H$ is isolated in a group $G$ if for every $x\in G$ we have either $x\in H$ or $\langle x\rangle\cap H=1$. In this short note, we describe the set of isolated subgroups of a finite abelian group. The technique used…

This work presents an example of a finitely presented semigroup $S$ containing an infinite nonnilpotent nil ideal $LS$, whose elements do not have a square (i.e. any word of the type $LXYYZ$ equals zero.)

Let $G$ be a finite group and assume $p$ is a prime dividing the order of $G$. Suppose for any such $p$, that every two abelian $p$-subgroups of $G$ of equal order are conjugate. The structure of such a group $G$ has been settled in this…

We classify the finite groups whose non-linear irreducible characters that are not conjugate under the natural Galois action have distinct degrees, therefore extending the results in Berkovich et al. [Proc. Amer. Math. Soc. {\bf 115}…

We give the first example of a non-linear residually finite 1-related group: < a, t | a^{t^2}=a^2>.

We prove that any finitely generated one ended group has linear end depth. Moreover, we give alternative proofs to theorems relating the growth of a finitely generated group to the number of its ends.

We provide the first examples of finitely generated simple groups that are amenable (and infinite). This follows from a general existence result on invariant states for piecewise-translations of the integers. The states are obtained by…

This survey purports to be an elementary introduction to compactly presented groups, which are the analogue of finitely presented groups in the broader realm of locally compact groups. In particular, compact presentation is interpreted as a…

We consider abelain subgroups of small index in finite groups. More generally, we consider subgroups such that the product of their index by the index of their centralizer is small.

We say that a finite group $G$ satisfies the independence property if, for every pair of distinct elements $x$ and $y$ of $G$, either $\{x,y\}$ is contained in a minimal generating set for $G$ or one of $x$ and $y$ is a power of the other.…

We construct first examples of infinite groups having property (T) whose Kazhdan constants admit a lower bound independent of the choice of a finite generating set.

Necessary and sufficient conditions for finite commutative semihypergroups to be built from abelian groups of the same order are established.