# Fixed points for group actions on 2-dimensional affine buildings

@article{Schillewaert2020FixedPF, title={Fixed points for group actions on 2-dimensional affine buildings}, author={Jeroen Schillewaert and Koen Struyve and Anne Thomas}, journal={arXiv: Group Theory}, year={2020} }

We prove a local-to-global result for fixed points of groups acting on affine buildings (possibly non-discrete) of types $\tilde{A}_1\times \tilde{A}_1, \tilde{A}_2$ or $\tilde{C}_2$. In the discrete case, our theorem establishes a conjecture by Marquis.

## References

SHOWING 1-10 OF 38 REFERENCES

### A fixed point theorem for Lie groups acting on buildings and applications to Kac–Moody theory

- Mathematics
- 2015

We establish a fixed point property for a certain class of locally compact groups, including almost connected Lie groups and compact groups of finite abelian width, which act by simplicial isometries…

### Torsion groups do not act on 2-dimensional CAT(0) complexes

- MathematicsDuke Mathematical Journal
- 2022

We show, under mild hypotheses, that if each element of a finitely generated group acting on a 2-dimensional CAT(0) complex has a fixed point, then the action is trivial. In particular, all actions…

### (Non-)completeness of R-buildings and fixed point theorems

- Mathematics
- 2009

We prove two generalizations of results proved by Bruhat and Tits involving metrical completeness and R-buildings. Firstly, we give a generalization of the Bruhat-Tits fixed point theorem also valid…

### At infinity of finite-dimensional CAT(0) spaces

- Mathematics
- 2010

We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space X has a non-empty intersection in the visual bordification $${ \overline{X} = X \cup \partial X}$$ .…

### A note on locally elliptic actions on cube complexes

- Mathematics
- 2018

We deduce from Sageev's results that whenever a group acts locally elliptically on a finite dimensional CAT(0) cube complex, then it must fix a point. As an application, we give an example of a group…

### Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings

- Mathematics
- 1997

Abstract We study quasi-isometries between products of symmetric spaces and Euclidean buildings. The main results are that quasi-isometries preserve the product structure, and that in the irreducible…

### On the joint spectral radius for isometries of non-positively curved spaces and uniform growth

- Mathematics
- 2018

We recast the notion of joint spectral radius in the setting of groups acting by isometries on non-positively curved spaces and give geometric versions of results of Berger-Wang and Bochi valid for…

### The Tits Alternative for Cat(0) Cubical Complexes

- Mathematics
- 2004

A Tits alternative theorem is proved in this paper for groups acting on CAT(0) cubical complexes. That is, a proof is given to show that if G is assumed to be a group for which there is a bound on…