Related papers: L^2-Betti numbers arising from the lamplighter gro…

An ideal $I \subset \mathbb{k}[x_1, \ldots, x_n]$ is said to have linear powers if $I^k$ has a linear minimal free resolution, for all $k$. In this paper we study the Betti numbers of $I^k$, for ideals $I$ with linear powers. The Betti…

We compute the Brauer group of the moduli stack of elliptic curves over the integers, localizations of the integers, finite fields of odd characteristic, and algebraically closed fields of characteristic not $2$. The methods involved…

For non-amenable finitely generated virtually free groups, we show that the combinatorial Euler characteristic introduced by Emerson and Meyer is the preimage of the K-theory class of higher Kazhdan projections under the Baum-Connes…

Let $G$ be an elementary abelian $p$-group. In this paper, we calculate the $K_2$-groups of some quotient rings $\mathbb{Z}[G]/I$ for certain ideals $I \subseteq \mathbb{Z}[G]$ of finite $p$-power index. These results are established…

We study a new class of codes over Z_2 x Z_2 which we call L-codes. They arise as a natural fifth step in a series of analogies between Kleinian codes, binary codes, lattices and vertex operator algebras. This analogy will be explained in…

We show that a finite type duality group of dimension $d>2$ is the fundamental group of a $(d+3)$-manifold with rationally acyclic universal cover. We use this to find closed manifolds with rationally acyclic universal cover and some…

Two conjectures about homology groups, K-groups and topological full groups of minimal etale groupoids on Cantor sets are formulated. We verify these conjectures for many examples of etale groupoids including products of etale groupoids…

A non-traditional proof of the Gregory-Leibniz series, based on the relationships among the zeta function, Bernoulli coefficients, and the Laurent expansion of the cotangent is given. New series for calculating pi are obtained.

We compute the Betti numbers of the geometric spaces associated to nonrational simple convex polytopes and find that they depend on the combinatorial type of the polytope exactly as in the rational case. This shows that the combinatorial…

We give a construction of homotopy algebras based on ``higher derived brackets''. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element $\Delta$. Given…

We determine the exact group structure of the abelianization of $\text{SL}_2(A)$, where $A$ is a Dedekind domain of arithmetic type with infinitely many units. In particular, our results show that $\text{SL}_2(A)^\text{ab}$ is finite, with…

We study $\ell^2$ Betti numbers, coherence, and virtual fibring of random groups in the few-relator model. In particular, random groups with negative Euler characteristic are coherent, have $\ell^2$ homology concentrated in dimension 1, and…

Motivated by the recent work of Bownik and Ross \cite{BR}, and Jakobsen and Lemvig \cite{JL}, this article generalizes latest results on reproducing formulas for generalized translation invariant (GTI) systems to the setting of super-spaces…

In this article, we study Lehmer-type bounds for the N\'eron-Tate height of $\bar{K}$-points on abelian varieties $A$ over number fields $K$. Then, we estimate the number of $K$-rational points on $A$ with N\'eron-Tate height $\leq \log B$…

Factoring out the spin $1$ subalgebra of a $ W $ algebra leads to a new $ W $ structure which can be seen either as a rational finitely generated $ W $ algebra or as a polynomial non-linear $ W_\infty$ realization.

In \cite{DJL07} it was shown that if $\scra$ is an affine hyperplane arrangement in $\C^n$, then at most one of the $L^2$--Betti numbers $b_i^{(2)}(\C^n\sm \scra,\id)$ is non--zero. In this note we prove an analogous statement for…

Two families $\mathcal A\subseteq\binom{[n]}{k}$ and $\mathcal B\subseteq\binom{[n]}{\ell}$ are called cross-$t$-intersecting if $|A\cap B|\geq t$ for all $A\in\mathcal A$, $B\in\mathcal B$. Let $n$, $k$ and $\ell$ be positive integers such…

For any numerical semigroup $S$, there are infinitely many numerical symmetric semigroups $T$ such that $S=\frac{T}{2}$ is their half. We are studying the Betti numbers of the numerical semigroup ring $K[T]$ when $S$ is a 3-generated…

A family of solvable self-dual Lie algebras is presented. There exist a few methods for the construction of non-reductive self-dual Lie algebras: an orthogonal direct product, a double-extension of an Abelian algebra, and a Wigner…

Using an application of Schmidt's Subspace Theorem, this paper gives new transcendence criteria for rapidly converging infinite products of algebraic numbers. The paper also improves existing criteria for irrationality of products and…