Related papers: Intersection-saturated groups without free subgrou…

We prove that for a stable theory $T,$ if $M$ is a saturated model of $T$ of cardinality $\lambda$ where $\lambda > \big|T\big|,$ then $Aut(M)$ has a dense free subgroup on $2^{\lambda}$ generators. This affirms a conjecture of Hodges.

A finitely generated group is lacunary hyperbolic if one of its asymptotic cones is an $\mathbb{R}$-tree. In this article we give a necessary and sufficient condition on lacunary hyperbolic groups in order to be stable under free product by…

Let $G$ be a group and $S$ be the set of all non-trivial proper subgroups of $G$. The intersection hypergraph of $G$, denoted by $\tilde{\Gamma}_\mathcal{H}(G)$, is a hypergraph whose vertex set is $\{H \in S \,\, | \,\, H \cap K = \{e\}…

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…

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph $G$, checks…

Let $G$ be a finite group and $S< G$. A cover for a group $G$ is a collection of subgroups of $G$ whose union is $G$. We use the term $n$-cover for a cover with $n$ members. A cover $\Pi =\{H_1, H_2, \dots, H_n\}$ is said to be a strict…

The minimum number of self-intersection points for members of a free homotopy class of curves on the punctured torus is bounded above in terms of the number L of letters required for a minimal description of the class in terms of the…

For a permutation group $G$ acting on a set $V$, a subset $\mathcal{F}$ of $G$ is said to be an intersecting set if for every pair of elements $g,h\in \mathcal{F}$ there exists $v \in V$ such that $g(v) = h(v)$. The intersection density…

Gray and Ruskuc have shown that any group G occurs as the maximal subgroup of some free idempotent generated semigroup IG(E) on a biordered set of idempotents E, thus resolving a long standing open question. Given the group G, they make a…

Graph $G$ is $F$-saturated if $G$ contains no copy of graph $F$ but any edge added to $G$ produces at least one copy of $F$. One common variant of saturation is to remove the former restriction: $G$ is $F$-semi-saturated if any edge added…

A subgroup $H$ of a finite group $G$ is said to satisfy $\Pi$-property in $G$ if for every chief factor $L/K$ of $G$, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi(HK/K\cap L/K)$-number. A subgroup $H$ of $G$ is called to be $\Pi$-supplemented in…

In this paper we study intersection configurations -- which describe the behaviour of multiple (finite) intersections of subgroups with respect to finite generability -- in the realm of free and free times free-abelian (FTFA) groups. We say…

Let G be a complex reductive algebraic group. We study complete intersections in a spherical homogeneous space G/H defined by a generic collection of sections from G-invariant linear systems. Whenever nonempty, all such complete…

Let $G$ be a group. Then $S\subseteq G$ is an invariable generating set of $G$ if every subset $S'$ obtained from $S$ by replacing each element with a conjugate is also a generating set of $G$. We investigate invariable generation among key…

The existence of an infinite simple boundedly generated 2-generated group and the existence of a boundedly simple 2-generated group containing a free non-cyclic subgroup are proved.

We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of…

We partly generalize the estimate for the rank of intersection of subgroups in free products of groups, proved earlier by S.V.Ivanov and W.Dicks, to the case of free amalgamated products of groups with normal finite amalgamated subgroup. We…

Every countable group $G$ can be embedded in a finitely generated group $G^*$ that is hopfian and complete, i.e. $G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is…

The residual closure of a subgroup $H$ of a group $G$ is the intersection of all virtually normal subgroups of $G$ containing $H$. We show that if $G$ is generated by finitely many cosets of $H$ and if $H$ is commensurated, then the…

We consider the topological full group of a substitution subshift induced by a substitution $a\to aca$, $b\to d$, $c\to b$, $d\to c$. This group is interesting since the Grigorchuk group naturally embeds into it. We show that the…