图小消去理论中的优化

图形小消去与边界作用的超有限性

Graphical small cancellation and hyperfiniteness of boundary actions

研究（无限表示的）图形小消去群在锥化Cayley图的Gromov边界上的作用，证明包括（无限表示的）经典小消去群在内的一类图形小消去群具有超有限边界作用。

Relative Hyperbolicity of Graphical Small Cancellation Groups

A piece of a labelled graph $Γ$ defined by D. Gruber is a labelled path that embeds into $Γ$ in two essentially different ways. We prove that graphical $Gr'(\frac{1}{6})$ small cancellation groups whose associated pieces have uniformly bounded length are relative hyperbolic. In fact, we show that the Cayley graph of such group presentation is asymptotically tree-graded with respect to the collection of all embedded components of the defining graph $Γ$, if and only if the pieces of $Γ$ are uniformly bounded. This implies the relative hyperbolicity by a result of C. Druţu, D. Osin and M. Sapir.

图论C(3)-T(6)蕴含CAT(0)

Graphical C(3)-T(6) implies CAT(0)

本文通过图论C(3)-T(6)复形构造局部CAT(0)度量，扩展了经典小取消理论的应用范围。

Rips-Segev torsion-free groups without the unique product property

We generalize the graphical small cancellation theory of Gromov to a graphical small cancellation theory over the free product. We extend Gromov's small cancellation theorem to the free product. We explain and generalize Rips-Segev's construction of torsion-free groups without the unique product property by viewing these groups as given by graphical small cancellation presentations over the free product. Our graphical small cancellation theorem then provides first examples of Gromov hyperbolic groups without the unique product property. We construct uncountably many non-isomorphic torsion-free…

Negative curvature in graphical small cancellation groups

We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical $Gr'(1/6)$ small cancellation groups. In particular, we characterize their 'contracting geodesics', which should be thought of as the geodesics that behave hyperbolically. We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group $G$ containing an element $g$ that is strongly contracting with respect to one finite generating set of $G$ …

Classifying spaces for families of subgroups for systolic groups

We determine the large scale geometry of the minimal displacement set of a hyperbolic isometry of a systolic complex. As a consequence, we describe the centraliser of such an isometry in a systolic group. Using these results, we construct a low-dimensional classifying space for the family of virtually cyclic subgroups of a group acting properly on a systolic complex. Its dimension coincides with the topological dimension of the complex if the latter is at least four. We show that graphical small cancellation complexes are classifying spaces for proper actions and that the groups acting on them…

Small cancellation labellings of some infinite graphs and applications

We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: - We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu's Property A) that are coarsely embeddable into a Hilbert space. Moreover, our groups act properly on CAT(0) cubical complexes. - We construct the first examples of finitely generated groups, with expanders …

Graphical complexes of groups

We introduce graphical complexes of groups, which can be thought of as a generalisation of Coxeter systems with 1-dimensional nerves. We show that these complexes are strictly developable, and we equip the resulting Basic Construction with three structures of non-positive curvature: piecewise linear CAT(0), C(6) graphical small cancellation, and a systolic one. We then use these structures to establish various properties of the fundamental groups of these complexes, such as biautomaticity and Tits Alternative. We isolate an easily checkable condition implying hyperbolicity of the fundamental g…

Infinitely presented C(6)-groups are SQ-universal

We prove that infinitely presented classical $C(6)$ small cancellation groups are SQ-universal. We extend the result to graphical $Gr_*(6)$-groups over free products. For every $p\in\mathbb{N}$, we construct uncountably many pairwise non-quasi-isometric groups that admit classical $C(p)$-presentations but no graphical $Gr'(\frac{1}{6})$-presentations.

Small cancellation theory over Burnside groups

We develop a version of small cancellation theory in the variety of Burnside groups. More precisely, we show that there exists a critical exponent $n_0$ such that for every odd integer $n\geq n_0$, the well-known classical $C'(1/6)$-small cancellation theory, as well as its graphical generalization and its version for free products, produce examples of infinite $n$-periodic groups. Our result gives a powerful tool for producing (uncountable collections of) examples of $n$-periodic groups with prescribed properties. It can be applied without any prior knowledge in the subject of $n$-periodic gr…

Residually finite non-exact groups

We construct the first examples of residually finite non-exact groups. The construction is based on author's earlier construction of groups containing isometrically expanders using a graphical small cancellation.