Combinatorial Modulus on Boundary of Right-Angled Hyperbolic Buildings

TL;DR

This paper explores the quasiconformal boundary structures of right-angled hyperbolic buildings using combinatorial methods, demonstrating the combinatorial Loewner property in certain 3- and 4-dimensional examples, which relates to rigidity results.

Contribution

It introduces the combinatorial Loewner property for boundaries of higher-dimensional right-angled hyperbolic buildings, extending previous work on 2-dimensional cases.

Findings

Boundaries of certain 3- and 4-dimensional buildings satisfy the combinatorial Loewner property.

The combinatorial Loewner property is a weaker version of the Loewner property.

This work suggests potential rigidity results for higher-dimensional hyperbolic buildings.

Abstract

In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner property. This is motivated by the fact that the quasiconformal structure of the boundary led to many results of rigidity in hyperbolic spaces since G.D. Mostow. In the case of buildings of dimension 2, many work have been done by M. Bourdon and H. Pajot. In particular, the Loewner property on the boundary permitted them to prove the quasi-isometry rigidity of right-angled Fuchsian buildings.