In $2011$, Osin and Thom conjectured that the first $\ell^2$-Betti number of a torsion-free discrete group is bounded above by the normal rank of the group minus one. The conjecture has surprising consequences for some fundamental problems in group theory and topology. These include the Wiegold problem on perfect groups, the Levin conjecture, the torsion-free case of the Kervaire conjecture, and an important special case of the Whitehead asphericity conjecture. In this article, we construct for each $n\in \mathbb{N}$ a countable torsion-free group $\Gamma_n$ such that $\beta^{(2)}_1(\Gamma_n)=n$ and so that the normal rank, $n(\Gamma_n)$, equals one. This disproves the conjecture. Our counterexamples are locally free and hence locally indicable. However, they are not finitely generated.
For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p. $$ Tang and Zhang conjectured an explicit formula for every finite $p>1$. We disprove the conjecture with two explicit real $2\times 2$ rank-one matrices at $p=3/2$. T...
We show that the weak Markus--Yamabe conjecture fails in every dimension $n\geq14$. We first prove a chain realization theorem: every polynomial Keller map $F=\I+H$ of $\R^n$ with component degrees $d_1,\ldots,d_n$ yields an explicit polynomial vector field on $\R^N$, with $N=\sum_i\max(d_i,2)-n$, whose Jacobian matrix...
Álvaro Castañeda, Gerardo Honorato, Francisco Valenzuela-Henríquez· 1 citation
We prove the generalized total rank conjecture over regular rings in characteristic $2$: if $R$ is a regular Noetherian domain of characteristic $2$ and $P$ is a differential $R$-module admitting a finite projective flag and having nonzero homology $H(P)$, then $\rank_R(P)\ge2^{\codim_RH(P)}$. In particular, we prove C...
Let $K$ be a global function field of characteristic $p>0$ and $\ell\neq p$ be a prime number. We study Selmer groups over a $\mathbb{Z}_\ell$-extension $K_\infty/K$. For a lisse $\mathbb Z_\ell$-sheaf we prove that the Pontryagin dual of the associated Selmer group is a finitely generated torsion module over the Iwasa...
We study the GIT quotients of the Peterson variety $\mathrm{Pet}_n\subset \mathrm{GL}(n,\mathbb C)/B$ under a one-parameter subgroup $\lambda:\mathbb G_m \to T$ with respect to the linearization $\mathcal L(\chi)$ given by a regular dominant character $\chi$ in the root lattice. Using the Richardson stratification, we...
Let $K/k$ be a finite abelian extension of number fields with Galois group $G$ and let $n\geq 2$. We prove, assuming the relevant $p$-part of the equivariant Tamagawa number conjecture, first-derivative analogues of the Deligne-Ribet integrality theorem and of the Coates-Sinnott conjecture. We construct a rank-one lead...
Saad El Boukhari· 0 citations
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