For Γ a relatively hyperbolic group, we construct a model for the universal space among Γ-spaces with isotropy on the family VC of virtually cyclic subgroups of Γ. We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in O+(n, 1) = Isom(Hn). We use the information we obtain to explicitly compute the lower algebraic K-theory of t...

The Polynomial Freiman-Ruzsa conjecture is one of the central open problems in additive combinatorics. It speculates tight quantitative bounds between combinatorial and algebraic notions of approximate subgroups. In this note, we restrict our attention to subsets of Euclidean space. In this regime, the original conjecture considers approximate algebraic subgroups as the set of lattice points in...

Derivative lattices associated with partially ordered linear spaces over skew fields are considered. Properties of the convex projective geometry $$ \mathcal{L} for a space FV field F investigated. The convexity subspaces means Abelian (ab-convexity), which is based on definition subgroup group. It shown that ab-convex directed plays theory same role as subgroups groups. We obtain element-wise ...

In this paper we classify fuzzy subgroups of a rank-3 abelian group $G = mathbb{Z}_{p^n} + mathbb{Z}_p + mathbb{Z}_p$ for any fixed prime $p$ and any positive integer $n$, using a natural equivalence relation given in cite{mur:01}. We present and prove explicit polynomial formulae for the number of (i) subgroups, (ii) maximal chains of subgroups, (iii) distinct fuzzy subgroups, (iv) non-isomorp...

Efficient discretisations of gauge groups are crucial with the long term perspective using tensor networks or quantum computers for lattice theory simulations. For any Lie group other than U$(1)$, however, there is no class asymptotically dense discrete subgroups. Therefore, limited to subgroups bound lead a freezing Monte Carlo simulations at weak couplings, necessitating alternative partition...

In this paper we classify fuzzy subgroups of the dihedral group $D_{pqrs}$  for distinct primes  $p$, $q$, $r$ and $s$. This follows similar work we have done on distinct fuzzy subgroups of some dihedral groups.We present formulae for the number of (i) distinct maximal chains of subgroups, (ii) distinct fuzzy subgroups and (iii) non-isomorphic classes of fuzzy subgroups under our chosen equival...

the starting point of this paper is given by priestley’s papers, where a theory of representation of distributive lattices is presented. the purpose of this paper is to develop a representation theory of fuzzy distributive lattices in the finite case. in this way, some results of priestley’s papers are extended. in the main theorem, we show that the category of finite fuzzy priestley space...

Abstract We show that the geometric realisation of poset proper parabolic subgroups a large-type Artin group has systolic geometry. use this geometry to set is stable under arbitrary intersections and forms lattice for inclusion. As an application, we groups are taking roots completely characterise conjugacy stable. also perspective recover unify results describing normalisers groups.

In this paper, we determine a presentation by explicit generators and relations for the Dade group of all (almost) extraspecial p-groups. The proof of the main result uses the cohomological properties of the Tits building corresponding to the natural geometric structure of the lattice of subgroups of such p-groups. AMS Subject Classification : 20C20, 20D15

For r = 7 and 8 we consider an action of the Weyl group of type E r on a unimodular lattice of rank r + 1. We give the tables of the first cohomology groups for all cyclic subgroups of the Weyl group with respect to this action. These are important in the arithmetic theory of Del Pezzo surfaces.