Related papers: On Wiegold's conjecture for the small Ree groups

Let $S=(a_1)\cdots(a_k)$ be a minimal zero-sum sequence over a finite cyclic group $G$. The index conjecture states that if $k=4$ and $\gcd(|G|,6)=1$, then $S$ has index 1. In this note we study the index conjecture and connect it to a…

In this note we prove the Weinstein conjecture for a class of symplectic manifolds including the uniruled manifolds based on Liu-Tian's result.

We provide new conditions for the Strong Atiyah conjecture to lift to finite group extensions. In particular, we show cocompact special groups satisfy these conditions, so the Strong Atiyah conjecture holds for virtually cocompact special…

The Hodge conjecture is shown to hold for rationally connected fivefolds, or more generally for fivefolds for which the base of the maximal rationally connected fibration is at most 3 dimensional.

We prove that the union-closed sets conjecture is true for separating union-closed families $\mathcal{A}$ with $|\mathcal{A}| \leq 2\left(m+\frac{m}{\log_2(m)-\log_2\log_2(m)}\right)$ where $m$ denotes the number of elements in…

In this paper we prove that Broue's abelian defect group conjecture holds for the Tits group $^2F_4(2)'$. Also we prove that under certain conditions we are able to lift derived equivalences and use this to prove Broue's conjecture for the…

We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to a sporadic simple group.

We generalize the basic results of Vinberg's \theta-groups, or periodically graded reductive Lie algebras, to fields of good positive characteristic. To this end we clarify the relationship between the little Weyl group and the (standard)…

For a group G relatively hyperbolic to a family of residually finite groups satisfying the Farrell-Jones conjecture, we reduce the solution of the Farrell-Jones conjecture for G to the case of certain nice cyclic extensions in G.

We present an application of Hodge theory towards the study of irreducible unitary representations of reductive Lie groups. We describe a conjecture about such representations and discuss some progress towards its proof.

Tate's theorem (Invent. Math. 1966)implies that the Tate conjecture holds for any abelian variety over a finite field whose Q_l-algebra of Tate classes is generated by those of degree 1. We construct families of abelian varieties over…

The present article gives the index formula of the principal congruence subgroups of the Hecke group H_5

The Huneke-Wiegand conjecture is a decades-long open question in commutative algebra. Garc\'ia-S\'anchez and Leamer showed that a special case of this conjecture concerning numerical semigroup rings $\Bbbk[\Gamma]$ can be answered in the…

Let $k\geqslant 3$ and let $A=\{0=a_{0}<a_{1}<\cdots<a_{k-1}\}$ with $\gcd(A)=1$. Freiman-Lev conjecture [V.F. Lev, Restricted set addition in groups, I. The classical setting, J. London Math. Soc. 62(2000), 27-40] is a well-known…

Abstract. We address the conjecture which states that an intersection of parabolic subgroups of an Artin-Tits group is a parabolic subgroup. We prove that the conjecture is equivalent to a, a priori, weaker conjecture. We also prove the…

A famous result of P. Heged\"us describes the structure of finite rational $\{2,5\}$-groups. In this paper, we show how Heged\"us' proof can be used to obtain a similar result for the more general situation of a finite rational $2$-group…

Several results about the union-closed sets conjecture are presented.

Let $G$ be a semisimple algebraic group over a field of characteristic $p > 0$. We prove that the dual Weyl modules for $G$ all have $p$-filtrations when $p$ is not too small. Moreover, we give applications of this theorem to…

Fuglede's spectral set conjecture states that a subset $\Omega$ of a locally compact abelian group $G$ tiles the group by translation if and only if there exists a subset of continuous group characters which is an orthogonal basis of…

The relative Novikov conjecture states that the relative higher signatures of manifolds with boundary are invariant under orientation-preserving homotopy equivalences of pairs. In this paper, we study the relative Baum-Connes assembly map…