The automorphism group is one of the most natural groups that acts on polytopes, since it captures its level of symmetry. The connection group is another group that acts naturally on the flags of a polytope; it can be interpreted as a recipe to recover the structure of the polytope from its flags. The chirality index is a measure of how far a chiral polytope is from being regular, and it is linked to a group called the chirality group. This chapter addresses these three groups and their interactions.

There is a canonical and efficient way to extend a convergent presentation of a category by a 2-polygraph into a coherent one. Precisely, the 3-cells used in this extension procedure are in one-to-one correspondence with the confluence diagrams of critical branchings in the polygraph. Now, if the polygraph is finite, so is the set of its critical branchings, and therefore the set of 3-cells generating coherence can be taken to be finite. In such a situation, the polygraph is said to have finite derivation type, or FDT. The relevance of this concept, introduced by Squier, lies in the following invariance property: if a category admits a finite presentation having finite derivation type, then all finite presentations of also have FDT. This invariance will prove essential to show that some finitely presented categories do not admit convergent presentations. Using these conditions, Squier managed to produce an explicit example of a finitely presented monoid, with decidable word problem, but having no finite convergent presentation. This provides a negative answer to the question of universality of finite convergent rewriting.

This chapter presents techniques for proving the termination of 3-polygraphs. A first method is based on a certain type of well-founded orders called reduction orders. Attention then turns to functorial interpretations: these amount to construct a functor from the underlying category to another category which already bears a reduction order. This covers quite a few useful examples. To address more complex cases, a powerful technique, due to Guiraud, is presented, based on the construction of a derivation from the polygraph. Here, termination is obtained by specifying quantities on 2-cells which decrease during rewriting, based on information propagated by the 2-cells themselves.

The study of universal algebra, that is, the description of algebraic structures by means of symbolic expressions subject to equations, dates back to the end of the 19th century. It was motivated by the large number of fundamental mathematical structures fitting into this framework: groups, rings, lattices, and so on. From the 1970s on, the algorithmic aspect became prominent and led to the notion of term rewriting system. This chapter briefly revisits these ideas from a polygraphic viewpoint, introducing only what is strictly necessary for understanding. Term rewriting systems are introduced as presentations of Lawvere theories, which are particular cartesian categories. It is shown that a term rewriting system can also be described by a 3-polygraph in which variables are handled explicitly, i.e., by taking into account their duplication and erasure. Finally, a precise meaning is given to the statement that term rewriting systems are "cartesian polygraphs".