# Borel structurability by locally finite simplicial complexes

## TL;DR

This paper demonstrates that countable Borel equivalence relations can be embedded into ones structurable by finite-dimensional contractible simplicial complexes with controlled vertex degrees, generalizing previous results for the case n=1.

## Contribution

It extends Jackson-Kechris-Louveau's result to higher dimensions, showing embeddings with bounded vertex degrees for structurable equivalence relations.

## Key findings

- Every countable Borel equivalence relation structurable by n-dimensional complexes embeds into one with bounded vertex degree.
- The bound on vertex degree is explicitly given by a formula involving n.
- The proof leverages classical Whitehead results on countable CW-complexes.

## Abstract

We show that every countable Borel equivalence relation structurable by $n$-dimensional contractible simplicial complexes embeds into one which is structurable by such complexes with the further property that each vertex belongs to at most $M_n := 2^{n-1}(n^2+3n+2)-2$ edges; this generalizes a result of Jackson-Kechris-Louveau in the case $n = 1$. The proof is based on that of a classical result of Whitehead on countable CW-complexes.