Geodesic
Contact graphs, boundaries, and a central limit theorem for $\mathrm{CAT}(0)$ cubical complexes
Talia Fernós1 orcid ; Jean Lécureux2 orcid ; Frédéric Mathéus3
1University of North Carolina at Greensboro, Greensboro, USA
2Université Paris-Saclay, Orsay, France
3Université de Bretagne Sud, Vannes, France
Groups, geometry, and dynamics, Tome 18 (2024) no. 2, pp. 677-704

Voir la notice de l'article provenant de la source EMS Press

DOI

Let X be a nonelementary CAT(0) cubical complex. We prove that if X is essential and irreducible, then the contact graph of X (introduced by Hagen (2014)) is unbounded and its boundary is homeomorphic to the regular boundary of X (defined by Fernós (2018) and Kar–Sageev (2016)). Using this, we reformulate the Caprace–Sageev’s rank-rigidity theorem in terms of the action on the contact graph. Let G be a group with a nonelementary action on X, and let (Zn​) be a random walk corresponding to a generating probability measure on G with finite second moment. Using this identification of the boundary of the contact graph, we prove a central limit theorem for (Zn​), namely that n​d(Zn​o,o)−nA​ converges in law to a non-degenerate Gaussian distribution (A=limn→∞​nd(Zn​o,o)​ is the drift of the random walk, and o∈X is an arbitrary basepoint).
DOI : 10.4171/ggd/775
Classification : 20F67, 60F05
Mots-clés : CAT(0) cube complex, random walk, contact graph, central limit theorem

Talia Fernós  1   ; Jean Lécureux  2   ; Frédéric Mathéus  3

1 University of North Carolina at Greensboro, Greensboro, USA
2 Université Paris-Saclay, Orsay, France
3 Université de Bretagne Sud, Vannes, France 
Talia Fernós; Jean Lécureux; Frédéric Mathéus. Contact graphs, boundaries, and a central limit theorem for $\mathrm{CAT}(0)$ cubical complexes. Groups, geometry, and dynamics, Tome 18 (2024) no. 2, pp. 677-704. doi: 10.4171/ggd/775
@article{10_4171_ggd_775,
     author = {Talia Fern\'os and Jean L\'ecureux and Fr\'ed\'eric Math\'eus},
     title = {Contact graphs, boundaries, and a central limit theorem for $\mathrm{CAT}(0)$ cubical complexes},
     journal = {Groups, geometry, and dynamics},
     pages = {677--704},
     year = {2024},
     volume = {18},
     number = {2},
     doi = {10.4171/ggd/775},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/775/}
}
TY  - JOUR
AU  - Talia Fernós
AU  - Jean Lécureux
AU  - Frédéric Mathéus
TI  - Contact graphs, boundaries, and a central limit theorem for $\mathrm{CAT}(0)$ cubical complexes
JO  - Groups, geometry, and dynamics
PY  - 2024
SP  - 677
EP  - 704
VL  - 18
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4171/ggd/775/
DO  - 10.4171/ggd/775
ID  - 10_4171_ggd_775
ER  - 
%0 Journal Article
%A Talia Fernós
%A Jean Lécureux
%A Frédéric Mathéus
%T Contact graphs, boundaries, and a central limit theorem for $\mathrm{CAT}(0)$ cubical complexes
%J Groups, geometry, and dynamics
%D 2024
%P 677-704
%V 18
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/775/
%R 10.4171/ggd/775
%F 10_4171_ggd_775

Cité par Sources :