We study actions of finitely generated groups on $\bbR$-trees under some stability hypotheses. We prove that either the group splits over some controlled subgroup (fixing an arc in particular), or the action can be obtained by gluing together actions of simple types: actions on simplicial trees, actions on lines, and actions coming from measured foliations on 2-orbifolds. This extends results by Sela and Rips-Sela. However, their results are misstated, and we give a counterexample to their statements.
The proof relies on an extended version of Scott's Lemma of independent interest. This statement claims that if a group $G$ is a direct limit of groups having suitably compatible splittings, then $G$ splits.
It is shown that for any action of a finitely presented group $G$ on an $\R$-tree, there is a decomposition of $G$ as the fundamental group of a graph of groups related to this action. If the action of $G$ on $T$ is non-trivial, i.e. there is no global fixed point, then $G$ has a non-trivial action on a simplcial $\R $-tree.
In his work on the Farrell-Jones Conjecture, Arthur Bartels introduced the concept of a "finitely $\mathcal{F}$-amenable" group action, where $\mathcal{F}$ is a family of subgroups. We show how a finitely $\mathcal{F}$-amenable action of a countable group $G$ on a compact metric space, where the asymptotic dimensions of the elements of $\mathcal{F}$ are bounded from above, gives an upper bound for the asymptotic dimension of $G$ viewed as a metric space with a proper left invariant metric. We generalize this to families $\mathcal{F}$ whose elements are contained in a collection, $\mathfrak{C}$…
We prove that a finitely generated pro-$p$ group $G$ acting on a pro-$p$ tree $T$ splits as a free amalgamated pro-$p$ product or a pro-$p$ HNN-extension over an edge stabilizer. If $G$ acts with finitely many vertex stabilizers up to conjugation we show that it is the fundamental pro-$p$ group of a finite graph of pro-$p$ groups $(\cal G, Γ)$ with edge and vertex groups being stabilizers of certain vertices and edges of $T$ respectively. If edge stabilizers are procyclic, we give a bound on $Γ$ in terms of the minimal number of generators of $G$. We also give a criterion for a pro-$p$ group $…
We prove that the class of finitely presented inverse monoids whose Schützenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, the languages of their Schützenberger automata are context-free. On the other hand, we show that there is a finitely presented inverse monoid with hyperbolic Schützenberger graphs and an unsolvable word problem.
Controlled Connectivity for Semi-Direct Products Acting on Locally Finite Trees
在2003年Bieri和Geoghegan通过定义$Σ^1(ρ)$推广了Bieri-Neuman-Strebel不变量$Σ^1$，$ρ$是一个在适当的CAT（0）空间$M$上由有限生成群$G$的等距作用。在本文中，我们将展示如何自然和众所周知的Bass-Serre理论和覆盖空间理论之间的联系提供了一个框架，$Σ^1(ρ)$的计算时，$ρ$是一个由$G = B \rtimes A$，$A$的有限生成群，在一个局部有限的Bass-Serre树$T$为$A$的协紧作用。这个框架导致一个定理，该定理提供了在$Σ^1(ρ)$中包括端点或从其中排除端点的条件。当$A$是一个作用在其Cayley图上的生成自由群时，我们可以从更代数的角度重述这一定理，从而得到了作用在$Σ^1$上的一些一般结果.
The goal of this article is to exhibit two new families of finitely generated simple groups of homeomorphisms of $\mathbf{R}$. These families are strikingly different from existing families owing to the nature of their actions on $\mathbf{R}$, and exhibit surprising algebraic and dynamical features. In particular, one construction provides the first examples of finitely generated simple groups of homeomorphisms of the real line which also admit a minimal action by homeomorphisms on the circle. This provides new examples of finitely generated simple groups with infinite commutator width, and th…
We prove existence of the large deviation principle, with a proper convex rate function, for the distribution of the renormalized distance from the origin of a random walk on a free product of finitely generated groups. As a consequence, we derive the same principle for nearest-neighbour random walks on regular trees.
Centre for Mathematical Sciences, Lund University（隆德大学数学中心）；Department of Mathematics, University of the Basque Country UPV/EHU（巴斯克大学数学系）；School of Mathematical Sciences, University of Southampton（南安普顿大学数学科学学院）
本文研究群作用于根树上的虚拟 retract，证明有限生成分支群不具有 LR 性质，并揭示 LR 性质在后临界有限二次复多项式中与欧几里得 orbifold 的关系，同时提供新的 pro-2 群例子。
