Aspherical Relative Presentations All Over Again

Quasirationality and aspherical (pro-$p$)presentations

Quasirational (pro-$p$)presentations are studied. The difference between aspherical and quasirational presentations sheds some light on the Whitehead's conjecture. We confirm expectations O.V. Melnikov on existence of a proper class of presentations.

关于展示的无球性的一次应用

On One Application of Asphericity of Presentations

本文利用组合无球性展示的命题，直接证明了 Kulikova 关于群 $F/[M,N]$ 挠率的引理 3(a)，并证明了满足 RC 条件时该群扩展是无挠的。

Aspherical groups and manifolds with extreme properties

We prove that every finitely generated group with recursive aspherical presentation embeds into a group with finite aspherical presentation. This and several known facts about groups and manifolds imply that there exists a 4-dimensional closed aspherical manifold $M$ such that the fundamental group $π_1(M)$ coarsely contains an expander, and so it has infinite asymptotic dimension, is not coarsely embeddable into a Hilbert space, and does not satisfy the Baum-Connes conjecture with coefficients. Closed aspherical manifolds with any of these properties were previously unknown.

Asphericity of a length four relative group presentation

We consider the relative group presentation $\mathcal{P} = \langle G, \mathbf{x} | \mathbf{r} \rangle$ where $\mathbf{x} = \{ x \}$ and $\mathbf{r} = \{ xg_1 xg_2 xg_3 x^{-1} g_4 \}$. We show modulo a small number of exceptional cases exactly when $\mathcal{P}$ is aspherical. If $H = \langle g_1^{-1} g_2, g_1^{-1} g_3 g_1 , g_4 \rangle \leq G$ then the exceptional cases occur when $H$ is isomorphic to one of $C_5,C_6,C_8$ or $C_2 \times C_4$.

Asphericity of groups defined by graphs

A graph $Γ$ labelled by a set $S$ defines a group $G(Γ)$ whose generators are the set of labels $S$ and whose relations are all words which can be read on closed paths of this graph. We introduce the notion of aspherical graph and prove that such a graph defines an aspherical group presentation. This result generalizes a theorem of Dominik Gruber on graphs satisfying graphical $C(6)$-condition and also allows to get new graphical conditions of asphericity analogous to some classical conditions.

On the asphericity of LOT-presentations of groups

Let $U$ be an arbitrary word in letters $x_1^{\pm 1}, ..., x_m^{\pm 1}$ and $m \ge 2$. We prove that the group presentation $<x_1, ..., x_m \|\ U x_i U^{-1} = x_{i+1}, i=1,..., m-1>$ is aspherical. The proof is based upon prior partial results of A. Klyachko and the author on the asphericity of such presentations.