Markus Lohrey, Benjamin Steinberg
In this paper we show that the membership problems for finitely generated submonoids and for rational subsets are recursively equivalent for groups with two or more ends.
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We show that the membership problem in a finitely generated submonoid of a graph group (also called a right-angled Artin group or a free partially commutative group) is decidable if and only if the independence graph (commutation graph) is a transitive forest. As a consequence we obtain the first example of a finitely presented group with a decidable generalized word problem that does not have a decidable membership problem for finitely generated submonoids. We also show that the rational subset membership problem is decidable for a graph group if and only if the independence graph is a transi…
We study the second bounded cohomology of an amalgamated free product of groups, and an HNN extension of a group. As an application, we have a group with infinitely many ends has infinite dimensional second bounded cohomology.
We show that each of Thompson's groups F, T, and V have infinitely many ends relative to certain subgroups. We go on to show that T and V both have Serre's property FA, i.e., any action of T or V on a tree will have a fixed point. (The proof of the latter statement was originally due to Ken Brown, and our proof is based on his notes.)
We show that any finitely generated group $F$ with infinitely many ends is not a group of fractions of any finitely generated proper subsemigroup $P$, that is $F$ cannot be expressed as a product $P P^{-1}$. In particular this solves a conjecture of Navas in the positive. As a corollary we obtain a new proof of the fact that finitely generated free groups do not admit isolated left-invariant orderings.
This article takes the inspiration from two milestones in the study of non minimal actions of groups on the circle: Duminy's theorem about the number of ends of semi-exceptional leaves and Ghys' freeness result in analytic regularity. Our first result concerns groups of analytic diffeomorphisms with infinitely many ends: if the action is non expanding, then the group is virtually free. The second result is a Duminy's theorem for minimal codimension one foliations: either non expandable leaves have infinitely many ends, or the holonomy pseudogroup preserves a projective structure.
Finite factorization is detected by undermonoids
本文证明了对于可消交换幺半群，所有子幺半群都是有限分解幺半群当且仅当所有下幺半群都是有限分解幺半群。
We study the space of ends of groups. For a finitely generated group, this is a Cantor space as soon as it is infinite. In contrast, we show that for infinitely generated countable groups, it exhibits several behaviors. For instance, we show that for the free product Z*Q, it is a Cantor space, while for a free group of infinite rank, it is not metrizable. For arbitrary countable groups, we actually establish an alternative: the space of ends is either metrizable, or has a continuous map onto the Stone-Cech compactification of N. We also show that the space of ends of a countable group has a co…
Property $R_{\infty}$ for groups with infinitely many ends
本文证明了具有无限多个终点的可及群以及特定类型的相对超几何群具有性质 $R_{\infty}$，并指出在有限呈现群中判断该性质是不可行的。
We prove the pro-$p$ version of the Karras, Pietrowski, Solitar, Cohen and Scott result stating that a virtually free group acts on a tree with finite vertex stabilizers. If a virtually free pro-$p$ group $G$ has finite centralizes of all non-trivial torsion elements more stronger statement is proved: $G$ embeds into a free pro-$p$ product of a free pro-$p$ group and finite $p$-group. The integral $p$-adic representation theory is used in the proof; it replaces the Stallings theory of ends in the pro-$p$ case.
The No-Core Principle for Stationary Actions and Ends of Stationary Random Subgroups
本文证明了可数群平稳作用的无核原理，并将其应用于平稳随机子群的几何，证明了其Schreier图几乎必然有0、1、2或无穷多个端，并构造了具有任意端数的回旋镖子群。
Self-avoiding walk is ballistic on graphs with more than one end
该研究证明在具有无限多个终点的图上，自回避行走表现出球形特性，并建立了自回避行走数量的渐近增长率。
In this paper we study obstructions to presentability by products for finitely generated groups. Along the way we develop both the concept of acentral subgroups, and the relations between presentability by products on the one hand, and certain geometric and measure or orbit equivalence invariants of groups on the other. This leads to many new examples of groups not presentable by products, including all groups with infinitely many ends, the (outer) automorphism groups of free groups, Thompson's groups, and even some elementary amenable groups.
On subgroups with narrow Schreier graphs
研究具有窄 Schreier 图的有限生成群对，证明虚拟纤维子群与无穷双重陪集的等价性，并给出群作用于 CAT(0) 立方复形时虚拟满射到整数的结论。
We introduce and study the operation, called dense amalgam, which to any tuple X_1,...,X_k of non-empty compact metric spaces associates some disconnected perfect compact metric space, denoted $\widetilde\sqcup(X_1,...,X_k)$, in which there are many appropriately distributed copies of the spaces X_1,...,X_k. We then show that, in various settings, the ideal boundary of the free product of groups is homeomorphic to the dense amalgam of boundaries of the factors. We give also related more general results for graphs of groups with finite edge groups. We justify these results by referring to a con…
A Hurewicz-type Theorem for the Dynamic Asymptotic Dimension with Applications to Coarse Geometry and Dynamics
本文证明动态渐近维数的Hurewicz型定理，并据此给出box空间渐近维数的上下界，应用于初等可解群作用及C*-代数分类。
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