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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…

For every $n \geq 1$, let $(\mathrm{FW}_n)$ denote the fixed-point property for median graphs of cubical dimension $n$ (or equivalently, for CAT(0) cube complexes of dimension $n$). In this article, we construct explicit examples of groups…

Stallings folding theory is modified, using double coset representatives, and to applied to the study of subgroups of amalgamated products of finite rank free groups. As a first application the subgroup membership problem for such groups is…

We provide new computations in bounded cohomology: A group is boundedly acyclic if its bounded cohomology with trivial real coefficients is zero in all positive degrees. We show that there exists a continuum of finitely generated…

We discuss in the context of finite extensions two classical theorems of Takahasi and Howson on subgroups of free groups. We provide bounds for the rank of the intersection of subgroups within classes of groups such as virtually free…

In this paper, using some properties of fundamental groups and covering spaces of connected polyhedra and CW-complexes, we present topological proof for some famous theorems about finitely presented groups.

In this article we will describe a finitely presented subgroup of Monod's group of piecewise projective homeomorphisms of R. This in particular provides a new example of a finitely presented group which is nonamenable and yet does not…

We prove that every finitely generated group with recursive aspherical presentation embeds into a group with finite aspherical presentation. This and several known facts about groups and manifolds imply that there exists a 4-dimensional…

We introduce the class of permawound unipotent groups, and show that they simultaneously satisfy certain "ubiquity" and "rigidity" properties that in combination render them very useful in the study of general wound unipotent groups. As an…

Let $S$ be a closed surface and $\text{Diff}_{\text{Vol}}(S)$ be the group of volume preserving diffeomorphisms of $S$. A finitely generated group $G$ is periodic of bounded exponent if there exists $k \in \mathbb{N}$ such that every…

We show that first-order formulae are concise in acylindrically hyperbolic groups and certain extensions thereof. We study further classes of groups, including Burnside groups, icc groups, groups with the `Big Powers' condition, torus knot…

We construct a finitely presented group with property (T) which can not act on on reasonable spaces. Such group is constructed using an generalization of Hall embedding theorem, where property (T) is added at the expense of weakening the…

Let G be a torsion-free abelian group of finite rank. The orbits of the action of Aut(G) on the set of maximal independent subsets of G determine the indecomposable decompositions of G. G contains a direct sum of pure strongly…

For each countable group $Q$ we produce a short exact sequence $1\to N \to G \to Q\to 1$ where $G$ is f.g. and has a graphical $\frac16$ presentation and $N$ is f.g. and satisfies property $T$. As a consequence we produce a group $N$ with…

In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with an excursion on projective structures on…

Unifying various constructions of quandles including Coxeter quandles, free quandles, knot quandles of prime knots and Dehn quandles of orientable surfaces, we introduce Dehn quandles of groups with respect to their subsets. It turns out…

Many bundle gerbes constructed in practice are either infinite-dimensional, or finite-dimensional but built using submersions that are far from being fibre bundles. Murray and Stevenson proved that gerbes on simply-connected manifolds,…

We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding…

In this note we look at presentations of subgroups of finitely presented groups with infinite cyclic quotients. We prove that if $H$ is a finitely generated normal subgroup of a finitely presented group $G$ with $G/H$ cyclic, then $H$ has…

Veech groups uniformize Teichm\"uller geodesic curves in Riemann moduli space. Recently, examples of infinitely generated Veech groups have been given. We show that these can even have infinitely many cusps and infinitely many infinite…