Related papers: Relating the Farrell Nil-groups to the Waldhausen …

We prove a natural generalization of Szep's conjecture. Given an almost simple group $G$ with socle not isomorphic to an orthogonal group having Witt defect zero, we classify all possible group elements $x,y\in G\setminus\{1\}$ with $G={\bf…

The Clifford group associated with a finite abelian group gives rise to a natural extension by the corresponding symplectic group. We prove that this extension splits as a semidirect product if and only if the group order is not divisible…

We investigate when Isomorphism Conjectures, such as the ones due to Baum-Connes, Bost and Farrell-Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that…

We show that the spectral set conjecture by Fuglede holds in the setting of cyclic groups of order $p^n q$, where $p$, $q$ are distinct primes and $n\geq1$. This means that a subset $E$ of such a group $G$ tiles the group by translation…

The inverse theorem for the Gowers norms, in the form proved by Green, Tao and Ziegler, applies to functions on an interval $[M]$. A recent paper of Candela and Sisask requires a stronger conclusion when applied to $N$-periodic functions;…

We investigate the indicators for certain groups of the form $\BZ_k\rtimes D_l$ and their doubles, where $D_l$ is the dihedral group of order $2l$. We subsequently obtain an infinite family of totally orthogonal, completely real groups…

The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group over a regular semilocal ring is itself trivial. Extending the work of \v{C}esnavi\v{c}ius and Fedorov, we prove a non-noetherian…

We introduce a group naturally acting on aperiodic necklaces of length $n$ with two colours using the 1--1 correspondences between aperiodic necklaces and irreducible polynomials over the field $\F_2$ of two elements. We notice that this…

We prove a surjectivity theorem for the Deligne canonical extension of a polarizable variation of Hodge structure with quasi-unipotent monodromy at infinity along the lines of Esnault-Viehweg. We deduce from it several injectivity theorems…

We prove an equivariant version of the fact that word-hyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the Farrell-Jones conjecture in algebraic K-theory for every…

This note is on spherical classes in $H_*(QS^0;k)$ when $k=\mathbb{Z},\mathbb{Z}/p$ with a special focus on the case of $p=2$ related to Curtis conjecture. We apply Freudenthal theorem to prove a vanishing result for the Hurewicz image of…

The notion of formal duality in finite Abelian groups appeared recently in relation to spherical designs, tight sphere packings, and energy minimizing configurations in Euclidean spaces. For finite cyclic groups it is conjectured that there…

On a cyclic group of prime order, the non-trivial Dirichlet characters together with their Fourier transforms have constant modulus outside 0 and vanish at 0. Answering a question of H. Cohn, we construct new functions with these…

Controlled $K$-theory is used to show that algebraic $K$-theory of virtually abelian groups is described by an assembly map defined using possibly-infinite hyperelementary subgroups. The Farrell-Jones summand (coming from infinite…

We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.

We show that finite-dimensional Lie algebras over a field of characteristic zero such that their high-degree cohomology in any finite-dimensional non-trivial irreducible module vanishes, are, essentially, direct sums of semisimple and…

Let A be a basic connected finite dimensional algebra over an algebraically closed field, let G be a group, let T be a basic tilting A-module and let B the endomorphism algebra of T. Under a hypothesis on T, we establish a correspondence…

We prove a generalization of the Conley conjecture: Every Hamiltonian diffeomorphism of a closed symplectic manifold has infinitely many periodic orbits if the first Chern class vanishes over the second fundamental group. In particular, we…

Generalizing a result of Furstenberg, we show that for every infinite discrete group $G$, the Bernoulli flow $2^G$ is disjoint from every minimal $G$-flow. From this, we deduce that the algebra generated by the minimal functions…

We give a survey of the meaning, status and applications of the Baum-Connes Conjecture about the topological K-theory of the reduced group C^*-algebra and the Farrell-Jones Conjecture about the algebraic K- and L-theory of the group ring of…