The rank gradient from a combinatorial viewpoint

Sublinearly Morse boundaries from the viewpoint of combinatorics

We prove that the sublinearly Morse boundary of every known cubulated group continuously injects in the Gromov boundary of a certain hyperbolic graph. We also show that for all CAT(0) cube complexes, convergence to sublinearly Morse geodesic rays has a simple combinatorial description using the hyperplanes crossed by such sequences. As an application of this combinatorial description, we show that a certain subspace of the Roller boundary continously surjects on the subspace of the visual boundary consisting of sublinearly Morse geodesic rays.

Rank, combinatorial cost and homology torsion growth in higher rank lattices

We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of right angled groups. A group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute. Most non-uniform lattices in higher rank simple Lie groups are right angled. We provide the first examples of uniform (co-co…

On the combinatorial and rank properties of certain subsemigroups of full contractions of a finite chain

Let $[n]=\{1,2,\ldots,n\}$ be a finite chain and let $\mathcal{CT}_{n}$ be the semigroup of full contractions on $[n]$. Denote $\mathcal{ORCT}_{n}$ and $\mathcal{OCT}_{n}$ to be the subsemigroup of order preserving or reversing and the subsemigroup of order preserving full contractions, respectively. It was shown in [17] that the collection of all regular elements (denoted by, Reg$(\mathcal{ORCT}_{n})$ and Reg$(\mathcal{OCT}_{n}$), respectively) and the collection of all idempotent elements (denoted by E$(\mathcal{ORCT}_{n})$ and E$(\mathcal{OCT}_{n}$), respectively) of the subsemigroups $\mat…

一种组合高阶双曲性条件

A combinatorial higher-rank hyperbolicity condition

本研究提出组合高阶双曲性即(n,δ)-双曲性的概念，证明其基础性质，并验证Helly群与分层双曲群可在对应空间上几何作用，拓展了双曲性理论。

秩二复反射群通过生成元与关系的一种组合推广

A Combinatorial Generalisation of Rank two Complex Reflection Groups via Generators and Relations

本文通过生成元与关系对秩二复反射群进行组合推广，提出 J-reflection groups 的一致展示，证明其中心为循环群，并完成反射同构分类。

The combinatorial cost

We study the combinatorial analogues of the classical invariants of measurable equivalence relations. We introduce the notion of cost and $β$-invariants (the analogue of the first $L^2$-Betti number introduced by Gaboriau) for sequences of finite graphs with uniformly bounded vertex degrees and examine the relation of these invariants and the rank gradient resp. mod $p$ homology gradient invariants introduced by Lackenby for residually finite groups.