# The configuration space of equidistant triples in the Heisenberg group

## TL;DR

This paper characterizes the space of all equidistant triples in the Heisenberg group with the Korányi metric, showing it forms a specific hypersurface in three-dimensional real space.

## Contribution

It establishes an explicit isomorphism between the configuration space of equidistant triples and a hypersurface in -dimensional real space, providing a geometric description.

## Key findings

- Configuration space is isomorphic to a hypersurface in D
- Provides a geometric description of equidistant triples
- Advances understanding of the Heisenberg group's metric geometry

## Abstract

We prove that the configuration space of equidistant triples on the Heisenberg group equipped with the Kor\'anyi metric, is isomorphic to a hypersurface of $\mathbb{R}^3$.