We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective and strongly shortcut spaces. show that every space admits a new is injective. The quasi-isometric to the original preserved under automorphisms space. uniformly bounded geometry shortcut. Consequently, groups -- including mapping class surfaces are Using these results, we deduce ...

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...

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...

We obtain a number of finiteness results for groups acting on hy-perbolic spaces and R-trees. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of n-generated one-ended subgroups. We also prove an acylindrical accessibility theorem for groups acting on R-trees.

Abstract We give a complete classification of finite subgroups automorphisms K3 surfaces up to deformation. The is in terms Hodge theoretic data associated certain conjugacy classes the orthogonal group lattice. moduli theory surfaces, particular surjectivity period map and strong Torelli theorem allow us interpret this datum geometrically. Our approach computer aided involves Hermitian lattice...

In the present paper, we study certain quotients of étale fundamental group a hyperbolic curve over field. We prove that action outer automorphism quotient an algebraically closed field on its conjugacy classes open subgroups is faithful. Also, that, if k either number or p-adic local field, then Galois representation associated to geometric ℙk1\{0,1,∞} injective.

Let G be a simple complex classical group and g its Lie algebra. Let U~(g) be the Drinfeld-Jimbo quantization of the universal enveloping algebra U(g). We construct an explicit U~(g)-equivariant quantization of conjugacy classes of G with Levi subgroups as the stabilizers.

COURSE CONTENT: Groups, Examples of groups, Some elementary properties of groups, Subgroups, Cyclic groups, Cosets and Langrange’s theorem, Normal subgroups and Quotient groups, Homomorphisms, The Isomorphism Theorems, Group Actions, orbits and stabilizers, Conjugacy, Class equation of a finite group, Cauchy’s Theorem, The structure of p-groups, The Sylow’s theorems, Some applications of Sylow’...