Deformation spaces of trees

JSJ decompositions: definitions, existence, uniqueness. II. Compatibility and acylindricity

This paper and its companion arXiv:0911.3173 have been replaced by arXiv:1602.05139. We define the compatibility JSJ tree of a group G over a class of subgroups. It exists whenever G is finitely presented and leads to a canonical tree (not a deformation space) which is invariant under automorphisms. Under acylindricity hypotheses, we prove that the (usual) JSJ deformation space and the compatibility JSJ tree exist, and we describe their flexible subgroups. We apply these results to finitely generated CSA groups, Γ-limit groups (allowing torsion), and relatively hyperbolic groups.

A Fixed Point Theorem for Deformation Spaces of G-trees

For a finitely generated free group F_n, of rank at least 2, any finite subgroup of Out(F_n) can be realized as a group of automorphisms of a graph with fundamental group F_n. This result, known as Out(F_n) realization, was proved by Zimmermann, Culler and Khramtsov. This theorem is comparable to Nielsen realization as proved by Kerckhoff: for a closed surface with negative Euler characteristic, any finite subgroup of the mapping class group can be realized as a group of isometries of a hyperbolic surface. Both of these theorems have restatements in terms of fixed points of actions on spaces n…

Trees of cylinders and canonical splittings

Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders T_c. This tree only depends on the deformation space of T; in particular, it is invariant under automorphisms of G if T is a JSJ splitting. We thus obtain Out(G)-invariant cyclic or abelian JSJ splittings. Furthermore, T_c has very strong compatibility properties (two trees are compatible if they have a…

A very short proof of Forester's rigidity result

The deformation space of a simplicial G-tree T is the set of G-trees which can be obtained from T by some collapse and expansion moves, or equivalently, which have the same elliptic subgroups as T. We give a short proof of a rigidity result by Forester which gives a sufficient condition for a deformation space to contain an Aut(G)-invariant G-tree. This gives a sufficient condition for a JSJ splitting to be invariant under automorphisms of G. More precisely, the theorem claims that a deformation space contains at most one strongly slide-free G-tree, where strongly slide-free means the followin…

JSJ decompositions of groups

This is an account of the theory of JSJ decompositions of finitely generated groups, as developed in the last twenty years or so. We give a simple general definition of JSJ decompositions (or rather of their Bass-Serre trees), as maximal universally elliptic trees. In general, there is no preferred JSJ decomposition, and the right object to consider is the whole set of JSJ decompositions, which forms a contractible space: the JSJ deformation space (analogous to Outer Space). We prove that JSJ decompositions exist for any finitely presented group, without any assumption on edge groups. When edg…

群的有限指数可达性

Finite-index accessibility of groups

本文证明了有限表示群在有限指数分裂下的可达性定理，并指出局部有限树（具有有限生成稳定子群）的变形空间在G-树的支配偏序中是极大的。

关于在曲面中嵌入圆的空间

On spaces of embeddings of circles in surfaces

该研究探讨了在非正曲率曲面中嵌入圆的空间结构，通过有限根树索引连通分支，并展示了其与编织自同构群的关系。

Embedding theorems for actions on generalized trees, I

Using suitable deformations of simplicial trees and the duality theory for median sets, we show that every free action on a median set can be extended to a free and transitive one. We also prove that the category of median groups is a reflective full subcategory of the category of free actions on pointed median sets.

Universal deformation rings and self-injective Nakayama algebras

Let $k$ be a field and let $Λ$ be an indecomposable finite dimensional $k$-algebra such that there is a stable equivalence of Morita type between $Λ$ and a self-injective split basic Nakayama algebra over $k$. We show that every indecomposable finitely generated $Λ$-module $V$ has a universal deformation ring $R(Λ,V)$ and we describe $R(Λ,V)$ explicitly as a quotient ring of a power series ring over $k$ in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to $p$-modular blocks of finite groups with cyclic defect groups.

李群胚作用的轨道固定形变空间

The orbit-fixing deformation spaces of an action of a Lie groupoid

本文通过模仿曲面Teichmüller空间重新表述了李群胚作用的形变空间，推广至可能非局部自由的作用与非紧流形，证明了上链刚性意味着形变空间为单点，并计算了$\mathop{\mathrm{PSL}}(2,\mathbb{R})$的具体形变空间。