Koji Fujiwara
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.
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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.
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.
Let G be a noncocompact irreducible arithmetic group over a global function field K of characteristic p, and let H be a finite-index, residually p-finite subgroup of G. We show that the cohomology of H in the dimension of its associated Euclidean building with coefficients in the field of p elements is infinite.
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。
It is well-known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group $G$ can be detected on the cohomology group $\mathrm{H}^1(G,R[G])$, where $R$ is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group $G$ the same information about the number of ends of $G$ in the sense of H. Abels can be provided by $\mathrm{dH}^1(G,\mathrm{Bi}(G))$, where $\mathrm{Bi}(G)$ is the rational discrete standard bimodule of…
Some invariants of totally disconnected locally compact groups: cohomology and combinatorics
本文利用 Coxeter 群的组合学方法，研究全不连通局部紧群的端数与有理离散上同调维数这两个不变量，建立它们与 building 的 Weyl 群及 flat-rank 的联系，并证明上同调维数为一时的树作用可及性结果。
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.
We show that if G is an infinitely generated locally (polycyclic-by-finite) group with cohomology almost everywhere finitary, then every finite subgroup of G acts freely and orthogonally on some sphere.
Let G be a group of type FP_n and let p>1. In this paper we show that the reduced l^p-homology of G is dual to the reduced l^q-cohomology for \frac{1}{p}+\frac{1}{q}=1. In our main theorem we show that for a group of type FP_n with a central element of infinite order the reduced l^p-cohomology vanishes. We generalize this fact for groups with infinitely many elements in the center of the group, for groups which are FCC, for groups with infinitely many finite conjugacy classes, for nilpotent groups, and for groups of polynomial growth.
Any action of a group $Γ$ on $\mathbb H^3$ by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to $Γ$. We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends of hyperbolic $3$-manifolds. That is, if two homotopy equivalent hyperbolic manifolds with infinite volume and without parabolic cusps have different geometrically infinite end invariants, then they define a $2$ dimensional subspace of bounded cohomology. Our techniques apply to…
Morse boundaries of graphs of groups with finite edge groups
本文证明自由积的莫尔斯边界只由因子的莫尔斯边界决定，并将该结论推广到具有有限边群和无穷多个端的群图，从而推广了Martin-Swiatkowski在非双曲群情形下的结果。
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。
A Manual for Ends, Semistability and Simple Connectivity at Infinity for Groups and Spaces
本文提供关于群和空间的终点、半稳定性及无穷远简单连通性的系统总结，涵盖最新研究成果和索引。
We study the geometry of infinitely presented groups satisfying the small cancelation condition C'(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C'(1/10). As a consequence, the Metric Approximation Property holds for the reduced C*-algebra and for the Fourier algebra of such groups. Our method further implies that the kernel of the comparison map between the bounded and the usual group cohomology in degree 2 has a basis of power…
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。
The homology groups introduced by A. Brumer can be used to establish a criterion ensuring that a profinite $\mathbb{F}_p[[G]]$-module of a pro-$p$ group $G$ has projective dimension $d<\infty$ (cf. Thm. A). This criterion yields a new characterization of free pro-$p$ groups (cf. Cor. B). Applied to a semi-direct factor $G\to\mathbb{Z}_p\to G$ isomorphic to $\mathbb{Z}_p$ which defines a non-trivial end in the sense of A.A. Korenev one concludes that the closure of the normal closure of the image of $σ$ is a free pro-$p$ subgroup (cf. Thm. C). From this result we will deduce a structure theorem…
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.
Elasticity of free type III actions of free groups
本研究证明保测度类的非顺从可树化III型等价关系可由任意生成元数的非阿贝尔自由群的自由作用诱导且对应Schreier图端不消失，借助已有III型刻画完成证明，凸显了III型与保测度情形的差异。
Let F_n be the free group on n generators, and PΣ_n be the group of automorphisms of F_n which send each generator to a conjugate of itself. Let K_n be the kernel of the homomorphism from PΣ_n to PΣ_{n-1} induced by mapping one of the free group generators to the identity. We show that K_n has cohomological dimension n-1, and that the ith cohomology groups are infinitely generated for all i between 2 and n-1. It follows that K_n is not finitely presentable for n>2.
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