S. V. Ivanov
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.
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We present a new test for studying asphericity and diagrammatic reducibility of group presentations. Our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails. We use this criterion to generalize results of J. Howie and S.M. Gersten on asphericity of LOTs and of Adian presentations, and derive new results on solvability of equations over groups. We also use our methods to investigate a conjecture of S.V. Ivanov related to Kaplansky's problem on zero divisors: we strengthen Ivanov's result for locally indicable groups and prove a weak version of t…
Asphericity of cubical presentations: the 2-dimensional case
本文研究二维$C(9)$立方表示，证明其锥化空间的二阶同伦群平凡，并给出相应基本群真作用的分类空间，进而在关系子非真幂条件下证明锥化空间是球面的。
The concept of asphericity for relative group presentations was introduced twenty five years ago. Since then, the subject has advanced and detailed asphericity classifications have been obtained for various families of one-relator relative presentations. Through this work the definition of asphericity has evolved and new applications have emerged. In this article we bring together key results on relative asphericity, update them, and exhibit them under a single set of definitions and terminology. We describe consequences of asphericity and present techniques for proving asphericity and for pro…
Two results on asphericity
该研究针对非球面性问题，构造了满足特定性质的非球面群表示复形链并回答了Hitchman的问题，还构造了具有关系间非平凡恒等式的积分李环表示，贡献了两个相关结果。
Asphericity of cubical presentations: the general case
本文证明 $C(9)$ 立方小消去表示对应锥化空间在适当条件下是球面的，进而为基本群构造分类空间，并应用于无 $\{3,4\}$ 标签的 Artin 群及 Wise 关于虚拟无挠性的问题。
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 $C_2 \times C_4$.
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。
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.
We prove that the generalised Fibonacci group F(r,n) is infinite for (r,n) in {(7 + 5k,5), (8 + 5k,5)} where k is greater than or equal to 0. This together with previously known results yields a complete classification of the finite F(r,n), a problem that has its origins in a question by J H Conway in 1965. The method is to show that a related relative presentation is aspherical from which it can be deduced that the groups are infinite.
For cyclically presented groups $G = G_n(w)$ with positive length four relators $w = x_0x_jx_kx_l$ in the free group with basis $x_0, x_1, \ldots, x_{n-1}$, we classify finiteness and, modulo two unresolved cases, we classify asphericity for the underlying presentations. We show that the fixed point subgroup of the shift $x_i \mapsto x_{i+1}$ is always finite and we relate finiteness of $G$ and asphericity to the dynamics of the shift action by the cyclic group of order $n$ on the nonidentity elements of $G$.
For group presentations with cyclic symmetry, there is a connection between asphericity and the dynamics of the shift automorphism. For the class of groups $G_n(k,l)$ described by the cyclic presentations $\mathcal{P}_n(k,l) = (x_i:x_ix_{i+k}x_{i+l}\ (i \mod n))$ and studied extensively by G. Williams and M. Edjvet \cite{EdjvetWilliams}, the shift acts freely on the nonidentity elements of $G_n(k,l)$ if and only if the presentation $\mathcal{P}_n(k,l)$ is combinatorially aspherical in the sense of \cite{CCH}. The shift has a nonidentity fixed point precisely when $G_n(k,l)$ is finite.
Local indicability of groups with homology circle presentations
本文给出具有圆同调呈示的群局部可指示的条件，并用于推广 LOT 群与非无圈 Adian 呈示的相关结论。
A relative one-relator presentation has the form P = < X,H ; R > where X is a set, H is a group, and R is a group word on X and H. We show that if the group word on X obtained from R by deleting all the terms from H has what we call the unique max-min property, then the group defined by P is residually finite if and only if H is residually finite (Theorem 1). We apply this to obtain new results concerning the residual finiteness of (ordinary) one-relator groups (Theorem 4). We also obtain results concerning the conjugacy problem for one-relator groups (Theorem 5), and results concerning the re…
To every semigroup presentation $\mathcal{P}= \langle Σ\mid \mathcal{R} \rangle$ and every baseword $w \in Σ^+$ can be associated a diagram group $D(\mathcal{P},w)$, defined as the fundamental group of the so-called Squier complex $S(\mathcal{P},w)$. Roughly speaking, $D(\mathcal{P},w)$ encodes the lack of asphericity of $\mathcal{P}$. Examples of diagram groups include Thompson's group $F$, the lamplighter group $\mathbb{Z} \wr \mathbb{Z}$, the pure planar braid groups, and various right-angled Artin groups. This survey aims at summarising what is known about the family of diagram groups.
We develop new computational methods for studying potential counterexamples to the Andrews-Curtis conjecture, in particular, Akbulut-Kurby examples AK(n). We devise a number of algorithms in an attempt to disprove the most interesting counterexample AK(3). To improve metric properties of the search space (which is a set of balanced presentations of the trivial group) we introduce a new transformation (called an ACM-move here) that generalizes the original Andrews-Curtis transformations and discuss details of a practical implementation. To reduce growth of the search space we introduce a strong…
Equations in Products of Free Groups and 3-Manifold Groups, I
本文构造了所有分裂坐标满射同态，给出了对应的平衡表示（包含所有闭可定向3-流形群），并分析了包括Olshanskii同态在内的一大类同态，证明分裂满射非常罕见且对应的平凡群平衡表示可通过Andrews-Curtis变换化为标准形式。
The problem of describing the group of units $\mathcal{U}(\mathbb{Z} G)$ of the integral group ring $\mathbb{Z} G$ of a finite group $G$ has attracted a lot of attention and providing presentations for such groups is a fundamental problem. Within the context of orders, a central problem is to describe a presentation of the unit group of an order $\mathcal{O}$ in the simple epimorphic images $A$ of the rational group algebra $\mathbb{Q} G$. Making use of the presentation part of Poincaré's Polyhedron Theorem, Pita, del Río and Ruiz proposed such a method for a large family of finite groups $G$ …
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