群论
A finite group $G$ is said to be semi-rational if the set of generators of each cyclic subgroup of $G$ is contained in at most two $G$-conjugacy classes. This is equivalent to the following condition: for every column of the character table…
We study centralizers of dilations in the quasi-isometry group of the positive real line. We introduce an asymptotic invariant defined via coarsely dense sequences at infinity and establish a rigidity theorem for quasi-isometries that…
We study doppelsemigroups, i.e., algebraic structures equip\-ped with two associative binary operations satisfying a specified system of axioms. We investigate duality and isomorphisms of doppelsemigroups and examine the relationships…
In the full transformation semigroup $T_n$ on a finite chain $X_n$, let $D_n=\{\alpha \in T_n:(\forall x \in X_n) \ x\alpha \leq x\}$ be the subsemigroup of all order-decreasing maps of $T_n$, and let $O_n=\{\alpha \in T_n:(\forall x ,y\in…
Classical works of Hall and McLain show that solubility and local nilpotency play a key role in deriving finite generation in groups from maximal or minimal conditions on normal subgroups. In this work, brace-theoretical analogues of Hall's…
We introduce the notion of sectional indecomposability and study it for finite groups: a group $H$ is sectionally indecomposable if, whenever $H$ is a section of a direct product $A \times B$, then $H$ is already a section of $A$ or of $B$.…
Reid--Smith parametrised ($P$)-closed groups acting on trees using graph-based combinatorial structures known as local action diagrams. Properties of the acting (topological) group, such as being locally compact, compactly generated,…
Let \(G<Aut(X)\) be a totally disconnected locally compact group acting strongly transitively on a locally finite building \(X\) of finite-rank and minimal non-spherical type. For sufficiently large thickness, every weakly mixing strongly…
We consider finite groups with at least three conjugacy class sizes that are composite numbers and we prove that, in that situation, the number of prime class sizes is bounded by the number of composite class sizes. The analogous result for…
Assume $n \geq 2$ and $\ell = (r_{1}, \ldots, r_{k}) \in [0,1]^{k}$ is an increasing sequence of real numbers. Let $G_{n,\ell}$ denote the group of orientation-preserving piecewise linear homeomorphisms $h$ of $I = [r_{1}, r_{k}]$ such…
We study the shadowing property for continuous endomorphisms of locally compact groups, using the left uniformity. For Lie groups we obtain a complete infinitesimal characterization: an endomorphism has shadowing if and only if its…
We show that under a suitable additional hypothesis the restricted Zassenhaus $\F_p$-Lie algebra or the rational Magnus Lie algebra of a free amalgamated product is the free amalgamated product of the corresponding Lie algebras of the…
We prove that a certain representation of the Baumslag-Gersten one-relator group $\mathrm{BG}(1,2)$ by germs of continuous functions is not faithful. This gives a negative answer to a problem of A. Yu. Olshanskii from 2010 (Problem 17.99 in…
We provide the first example of a finitely presented, and the first example of a simple, group of non-uniform exponential growth. The example is given by Thompson's group V.
Consider, on the space of marked groups, the map $\mathrm{Res}_{\mathcal{C}}$ which associates to a marked group its greatest residually-$\mathcal{C}$ quotient, for different sets $\mathcal{C}$ of groups. Except for trivial cases, this map…
We show that groups with a mild form of non-positive curvature (a navigable path system) satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element. This class includes discrete groups of projective…
The classes of abelian groups that are (uniformly) strongly Hopfian abelian groups, and dually, (uniformly) strongly co-Hopfian abelian groups have been studied by several authors, including Abdelalim (2015) and Abdelalim-Chillali-Essanouni…
We show that the affine cactus group is a CAT$(0)$ group for all degrees. Furthermore, we show that the affine cactus group $AJ_3$ of degree three is a hyperbolic group.
Affine Coxeter groups are fundamental objects in mathematics and in crystallography. If two group elements are conjugate, then they have very similar algebraic and geometric properties. Using recent structural results of Mili\'cevi\'c,…
Nagano spaces are compact symmetric spaces that admit large transformation groups. They include for instance all the Grassmannians and the Einstein Universes. In this paper, we study a Kobayashi-type pseudometric on domains in real-type…