论文标题
仿射结构，花圈产品和免费的仿射动作在线性非一切型树木上
Affine structures, wreath products and free affine actions on linear non-archimedean trees
论文作者
论文摘要
令$λ$为订购的Abelian集团，$ \ mathrm {aut}^+（λ）$ $λ$，$ g $的订单保留自动形态和$ a $ g $ a g $ a：g \ to \ to \ to \ to \ to \ mathrm {aut}^+（λ）$ a romomorphism y ryomorphismismism。在$λ$ -tree $ x $上$ g $的$α$ - 携带动作是满足$ d（gx，gy）=α_gd（x，y）$（$ x，y \ in x $，$ g \ in g $）的$λ$ -g $。我们考虑在$ x =λ$的情况下，可以承认免费，僵化的仿射动作的组类。此类组形成的类别比等轴测情况大得多。我们特别表明unitriangular组$ \ mathrm {ut}（n，\ m mathbb {r}）$和$ t^**（n，\ mathbb {r}）上的上三角矩阵的$ \ m m i \ mathbb {r} $，带有阳性的对角线，并带有阳性的对角线。我们的证明涉及各个谎言代数的左对称结构以及所讨论的组上相关的仿射结构。我们还表明，给定的订购的Abelian Groups $λ_0$和$λ_1$，以及$ G $ $ g $ $λ_0$上的定向式仿射动作，我们在合适的$λ'$上获得了花圈产品$ g \wrλ_1$的另一这样的动作。 因此，所有自由的可溶性组，剩余的群体和本地残留的无扭转nilpotent群体基本上承认在某些$λ'$上基本上免费的仿射动作。
Let $Λ$ be an ordered abelian group, $\mathrm{Aut}^+(Λ)$ the group of order-preserving automorphisms of $Λ$, $G$ a group and $α:G\to\mathrm{Aut}^+(Λ)$ a homomorphism. An $α$-affine action of $G$ on a $Λ$-tree $X$ is one that satisfies $d(gx,gy)=α_gd(x,y)$ ($x,y\in X$, $g\in G$). We consider classes of groups that admit a free, rigid, affine action in the case where $X=Λ$. Such groups form a much larger class than in the isometric case. We show in particular that unitriangular groups $\mathrm{UT}(n,\mathbb{R})$ and groups $T^*(n,\mathbb{R})$ of upper triangular matrices over $\mathbb{R}$ with positive diagonal entries admit free affine actions. Our proofs involve left symmetric structures on the respective Lie algebras and the associated affine structures on the groups in question. We also show that given ordered abelian groups $Λ_0$ and $Λ_1$ and an orientation-preserving affine action of $G$ on $Λ_0$, we obtain another such action of the wreath product $G\wr Λ_1$ on a suitable $Λ'$. It follows that all free soluble groups, residually free groups and locally residually torsion-free nilpotent groups admit essentially free affine actions on some $Λ'$.
