# Problems on automorphism groups of nonpositively curved polyhedral complexes and their lattices

@article{Farb2008ProblemsOA, title={Problems on automorphism groups of nonpositively curved polyhedral complexes and their lattices}, author={Benson Farb and G. Christopher Hruska and Anne Thomas}, journal={arXiv: Group Theory}, year={2008} }

The goal of this paper is to present a number of problems about automorphism groups of nonpositively curved polyhedral complexes and their lattices, meant to highlight possible directions for future research.

## 37 Citations

### Finiteness Properties of Non-Uniform Lattices on CAT(0) Polyhedral Complexes

- Mathematics
- 2011

We show that the homological finiteness length of a non-uniform lattice on a locally finite CAT(0) n-dimensional polyhedral complex is less than n. As a corollary, we obtain an upper bound for the…

### Automorphisms of geometric structures associated to Coxeter groups

- Mathematics
- 2012

In this paper, we consider the automorphism groups of the Cayley graph with respect to the Coxeter generators and the Davis complex of an arbitrary Coxeter group. We determine for which Coxeter…

### Isometry groups and lattices of non-positively curved spaces

- Mathematics
- 2008

We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces and their discrete subgroups. Classical results on Hadamard manifolds are extended to…

### Isometry groups of non‐positively curved spaces: discrete subgroups

- Mathematics
- 2009

We study lattices in non‐positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided.…

### Uniqueness of homogeneous CAT(0) polygonal complexes

- Mathematics
- 2014

We provide a combinatorial condition on a finite connected graph, $$L$$, for which there exists a unique CAT(0) polygonal complex such that the link at each vertex is $$L$$. Under the further…

### Automorphism Groups of Combinatorial Structures

- Mathematics
- 2018

This is a series of lecture notes taken by students during a five lecture series presented by Anne Thomas in 2016 at the MATRIX workshop: The Winter of Disconnectedness.

### Buildings, Group Homology and Lattices

- Mathematics
- 2010

This is the author's PhD thesis, published at the Universit\"at M\"unster, Germany in 2010. It contains a detailed description of the results of arXiv:0903.1989, arXiv:0905.0071 and arXiv:0908.2713.

### Tits Alternative for 2-dimensional CAT(0) complexes

- Mathematics
- 2021

We prove the Tits Alternative for groups acting on 2-dimensional CAT(0) complexes with a bound on the order of the cell stabilisers.

### On regular CAT(0) cube complexes and the simplicity of automorphism groups of rank-one CAT(0) cube complexes

- Mathematics
- 2018

We provide a necessary and sufficient condition on a finite flag simplicial complex, L, for which there exists a unique CAT(0) cube complex whose vertex links are all isomorphic to L. We then find…

### Commensurating HNN extensions: nonpositive curvature and biautomaticity

- MathematicsGeometry & Topology
- 2021

We show that the commensurator of any quasiconvex abelian subgroup in a biautomatic group is small, in the sense that it has finite image in the abstract commensurator of the subgroup. Using this…

## References

SHOWING 1-10 OF 126 REFERENCES

### Polygonal complexes and combinatorial group theory

- Mathematics
- 1994

We study the structure of certain simply connected 2-dimensional complexes with non-positive curvature. We obtain a precise description of how these complexes behave at infinity and prove an…

### On L2-cohomology and Property (T) for Automorphism Groups of Polyhedral Cell Complexes

- Mathematics
- 1997

Abstract. We present an update of Garland's work on the cohomology of certain groups, construct a class of groups many of which satisfy Kazhdan's Property (T) and show that properly discontinuous and…

### Covolumes of uniform lattices acting on polyhedral complexes

- Mathematics
- 2005

Let X be a polyhedral complex with finitely many isometry classes of links. We establish a restriction on the covolumes of uniform lattices acting on X. When X is two‐dimensional and has all links…

### COUNTING OVERLATTICES FOR POLYHEDRAL COMPLEXES

- Mathematics
- 2010

We investigate the asymptotics of the number of \overlattices" of a cocompact lattice in Aut( X), where X is a locally nite polyhedral complex. We use complexes of groups to prove an upper bound for…

### Regular Path Systems and (Bi)automatic Groups

- Mathematics
- 2006

We develop a new method of proving that groups acting on various classes of spaces, with nontrivial stabilizers admitted, are automatic or biautomatic. We apply this method to buildings, CAT(0)…

### Finite index subgroups of graph products

- Mathematics
- 2008

We prove that every quasiconvex subgroup of a right-angled Coxeter group is an intersection of finite index subgroups. From this we deduce similar separability results for other types of groups,…

### Counting Overlattices in Automorphism Groups of Trees

- Mathematics
- 2005

We give an upper bound for the number uΓ(n) of “overlattices” in the automorphism group of a tree, containing a fixed lattice Γ with index n. For an example of Γ in the automorphism group of a…

### Combinatorial structure of some hyperbolic buildings

- Mathematics
- 2002

Abstract. We give an elementary construction of polyhedra whose links are connected bipartite graphs. In particular, we construct polyhedra whose links are generalized m-gons. The polyhedra of this…