Related papers: Strong accessibility of Coxeter groups over minima…

The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible…

We give two criteria for diagrams of Heegaard splittings of 3-manifolds. Weaker one of them guarantees that the splitting is strongly irreducible, and the stronger one guarantees in addition that the Goeritz group is finite. They are…

For a simple graph $G=(V,E)$ without any isolated vertex, a cosecure dominating set $D$ of $G$ satisfies the following two properties (i) $S$ is a dominating set of $G$, (ii) for every vertex $v \in S$ there exists a vertex $u \in V…

We show that the Right-Angled Coxeter group $C=C(G)$ associated to a random graph $G\sim \mathcal{G}(n,p)$ with $\frac{\log n + \log\log n + \omega(1)}{n} \leq p < 1- \omega(n^{-2})$ virtually algebraically fibers. This means that $C$ has a…

The idea that the cohomology of finite groups might be fruitfully approached via the cohomology of ambient semisimple algebraic groups was first shown to be viable in the papers [CPS75] and [CPSvdK77]. The second paper introduced, through a…

Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian. Our main result gives an explicit, computable "visual" construction of this tree for…

For a graph $G$, $k(G)$ denotes its connectivity. A graph is super connected if every minimum vertex-cut isolates a vertex. Also $k_{1}$-connectivity of a connected graph is the minimum number of vertices whose deletion gives a disconnected…

We prove that a non-spherical irreducible Coxeter group is (directly) indecomposable and that a non-spherical and non-affine Coxeter group is strongly indecomposable in the sense that all its finite index subgroups are (directly)…

In this article, we study the outer automorphism group of a group G decomposed as a finite graph of group with finite edge groups and finitely generated vertex groups with at most one end. We show that Out(G) is essentially obtained by…

We show that the standard generating set of a Coxeter group is of minimal cardinality provided that the non-diagonal entries of the Coxeter matrix are sufficiently large.

We study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $\cE = \ZZ[e^{2 \pi i/3}]$: there are only four such lattices,…

In this paper we continue the study of prime graphs of finite solvable groups. The prime graph, or Gruenberg-Kegel graph, of a finite group G has vertices consisting of the prime divisors of the order of G and an edge from primes p to q if…

A graph $X$ is defined inductively to be $(a_0,\dots,a_{n-1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius $1$ around $v$ is an $(a_1,\dots,a_{n-1})$-regular graph. Such a graph $X$ is said to be…

In this paper we study the commensurability of hyperbolic Coxeter groups of finite covolume, providing three necessary conditions for commensurability. Moreover we tackle different topics around the field of definition of a hyperbolic…

A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it…

In this paper, we classify all the finite groups $G$ such that the commuting graph $\Gamma_C(G)$, order-sum graph $\Gamma_{OS}(G)$ and non-inverse graph $\Gamma_{NI}(G)$ are minimally edge connected graphs. We also classify all the finite…

We first prove that for every vertex x of a 4-connected graph G there exists a subgraph H in G isomorphic to a subdivision of the complete graph K4 on four vertices such that G-V(H) is connected and contains x. This implies an affirmative…

The Chermak-Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups of $G$. The least element of the Chermak-Delgado lattice of $G$ is known as the Chermak-Delgado subgroup of $G$. This paper…

We study classes of right-angled Coxeter groups with respect to the strong submodel relation of parabolic subgroup. We show that the class of all right-angled Coxeter group is not smooth, and establish some general combinatorial criteria…

The Divisibility Graph of a finite group $G$ has vertex set the set of conjugacy class lengths of non-central elements in $G$ and two vertices are connected by an edge if one divides the other. We determine the connected components of the…