Diagram rigidity for geometric amalgamations of free groups

Rigidity property of diagram groups

We show a rigid connection between two classical objects: the R.Thompson group F and the Dunce hat. A diagram group of a directed 2-complex contains a copy of F if and only if the complex contains a copy of the Dunce hat.

Rigidity of two-dimensional Coxeter groups

A Coxeter system is called two-dimensional if its associated Davis complex is two-dimensional (equivalently, every spherical subgroup has rank less than or equal to 2). We prove that given a two-dimensional system (W,S) and any other system (W,S') which yields the same reflections, the diagrams corresponding to these systems are isomorphic, up to the operation of diagram twisting defined by N. Brady, J. McCammond, B. Muhlherr, and W. Neumann. As a step in the proof of this result, certain two-dimensional groups are shown to be reflection rigid in the sense defined by the above authors, and a r…

星形Coxeter系统对称性与刚性

Symmetry and Rigidity of Star-Shaped Coxeter Systems

本文研究了具有星形有限Coxeter图的Coxeter群的自同构群，证明了每个自同构可分解为内自同构、图自同构以及三种额外类型：转置和两个部分共轭家族，并探讨了自然的短正合列，利用Moussong的超几何性准则证明这些群具有R_∞性质，并利用已知技术建立了这些群的刚性性质，解决了星形Coxeter系统的同构问题。

A boundary version of Cartan-Hadamard and applications to rigidity

In this paper, we prove a version of the classical Cartan-Hadamard theorem for negatively curved manifolds, of dimension $n\neq 5$, with non-empty totally geodesic boundary. More precisely, if $M_1^n,M_2^n$ are any two such manifolds, we show that (1) $\partial ^\infty \tilde M_1^n$ is homeomorphic to $\partial ^\infty \tilde M_2^n$, and (2) $\tilde M_1^n$ is homeomorphic to $\tilde M_2^n$. As a sample application, we show that simple, thick, negatively curved P-manifolds of dimension $\geq 6$ are topologically rigid. We include some straightforward consequences of topological rigidity (diagra…

Diagram groups and directed 2-complexes: homotopy and homology

We show that diagram groups can be viewed as fundamental groups of spaces of positive paths on directed 2-complexes (these spaces of paths turn out to be classifying spaces). Thus diagram groups are analogs of second homotopy groups, although diagram groups are as a rule non-Abelian. Part of the paper is a review of the previous results from this point of view. In particular, we show that the so called rigidity of the R.Thompson's group $F$ and some other groups is similar to the flat torus theorem. We find several finitely presented diagram groups (even of type F_\infty) each of which contain…

The Dynkin diagram cohomology of finite Coxeter groups

Let D be a connected graph. The Dynkin complex CD(A) of a D-algebra A was introduced by the second author in [TL2] to control the deformations of quasi-Coxeter algebra structures on A. In the present paper, we study the cohomology of this complex when A is the group algebra of a Coxeter group W and D is the Dynkin diagram of W. We compute this cohomology when W is finite and prove in particular the rigidity of quasi-Coxeter algebra structures on kW. For an arbitrary W, we compute the top cohomology group and obtain a number of additional partial results when W is affine. Our computations are c…