Detecting topological properties of Markov compacta with combinatorial properties of their diagrams

TL;DR

This paper introduces a formalism to analyze the topological properties of Markov compacta using combinatorial diagrams, enabling the detection of complex topological features from discrete structures.

Contribution

It provides a novel method to determine topological properties of Markov compacta through combinatorial analysis of their diagrammatic building blocks.

Findings

Topological properties like $k$-connectedness can be detected from diagrams.

The formalism encodes complex spaces with finite combinatorial objects.

Applicable to boundaries of hyperbolic groups in geometric group theory.

Abstract

We develop a formalism that allows us to describe Markov compacta with finite sets of diagrams that are building blocks of the entire sequence. This encodes complex, continuous spaces with discrete collections of combinatorial objects. We show that topological properties of the limit (such as $k$-connectedness, local $k$-connectedness or the disjoint arcs property) may be detected by looking at combinatorial properties of the diagrams. Markov compacta were introduced by M. Gromov and were motivated by some examples in geometric group theory. In particular, boundaries at infinity of hyperbolic groups belong to this class.