# Telescopic groups and symmetries of combinatorial maps

## TL;DR

This paper demonstrates that a wide variety of combinatorial and topological objects can realize any finite automorphism group, providing new universal techniques and showing the abundance of such objects with given symmetries.

## Contribution

It introduces a universal method for realizing any finite automorphism group across multiple classes of combinatorial and topological objects.

## Key findings

- Any finite automorphism group can be realized by many non-isomorphic objects.
- The realization technique applies broadly to maps, hypermaps, and related structures.
- There are super-exponentially many objects with a given automorphism group.

## Abstract

In the present paper, we show that many combinatorial and topological objects, such as maps, hypermaps, three-dimensional pavings, constellations and branched coverings of the two--sphere admit any given finite automorphism group. This enhances the already known results by Frucht, Cori -- Mach\`i, \v{S}ir\'{a}\v{n} -- \v{S}koviera, and other authors. We also provide a more universal technique for showing that ``any finite automorphism group is possible'', that is applicable to wider classes or, in contrast, to more particular sub-classes of said combinatorial and geometric objects. Finally, we show that any given finite automorphism group can be realised by sufficiently many non-isomorphic such entities (super-exponentially many with respect to a certain combinatorial complexity measure).