Recognisably Context-Free Subsets of Finitely Generated Groups

Kernkonzepte

Finitely generated groups have subsets whose set of words representing the elements of the subset form a context-free language. These are called recognisably context-free sets, and they have interesting properties related to the structure of the group.

Zusammenfassung

The content discusses the concept of recognisably context-free subsets of finitely generated groups. Key points:

1. Recognisably context-free sets are subsets of a group where the set of words representing the elements of the subset form a context-free language. This property is independent of the choice of finite generating set for the group.

2. The class of recognisably context-free sets has several closure properties, such as being closed under multiplication by rational sets and intersection with recognisable sets.

3. The author shows that in virtually free groups, every conjugacy class is recognisably context-free. This characterizes the class of groups where all conjugacy classes are recognisably context-free.

4. For subgroups and cosets whose Schreier coset graph is quasi-transitive, the author shows that the subset is recognisably context-free if and only if the Schreier coset graph is quasi-isometric to a tree (a quasi-tree).

The content provides a detailed analysis of the structure and properties of recognisably context-free subsets in finitely generated groups, with a focus on virtually free groups and quasi-transitive subsets.

Statistiken

Every conjugacy class of a finitely generated group G is recognisably context-free if and only if G is virtually free.
A coset Hg of a subgroup H in a finitely generated group G is recognisably context-free if and only if the Schreier coset graph of (G, H) is quasi-isometric to a tree.

Zitate

"Finitely generated groups have subsets whose set of words representing the elements of the subset form a context-free language. These are called recognisably context-free sets, and they have interesting properties related to the structure of the group."
"In virtually free groups, every conjugacy class is recognisably context-free. This characterizes the class of groups where all conjugacy classes are recognisably context-free."
"For subgroups and cosets whose Schreier coset graph is quasi-transitive, the subset is recognisably context-free if and only if the Schreier coset graph is quasi-isometric to a tree (a quasi-tree)."

Wichtige Erkenntnisse aus

Subsets of groups with context-free preimages

by Alex Levine um arxiv.org 05-01-2024

https://arxiv.org/pdf/2312.04191.pdf

Tiefere Fragen

What other properties or characterizations of recognisably context-free subsets in finitely generated groups are known

Recognisably context-free subsets in finitely generated groups have several other important properties and characterizations. One key property is that recognisably context-free sets are closed under certain operations. For example, the intersection of two recognisably context-free sets is also recognisably context-free. This closure property allows for the manipulation and combination of recognisably context-free sets in a way that preserves their recognisably context-free nature. Additionally, recognisably context-free sets are stable under preimages of free monoid homomorphisms, meaning that if a set is recognisably context-free in one group, its preimage under a homomorphism to another group is also recognisably context-free.

Are there any connections between the structure of a group's Cayley graph and the recognisably context-free subsets of the group

There is a significant connection between the structure of a group's Cayley graph and the recognisably context-free subsets of the group. The Cayley graph of a group provides a visual representation of the group's structure based on its generating set. In the context of recognisably context-free subsets, the Cayley graph can help identify patterns and relationships within the group that correspond to recognisably context-free languages. For example, if a group's Cayley graph is quasi-isometric to a tree, it indicates that the group has recognisably context-free subsets. This connection between the geometric properties of the Cayley graph and the formal language properties of the group highlights the interplay between group theory and formal language theory.

How do the results on recognisably context-free subsets relate to the study of the word problem and other formal language properties of groups

The results on recognisably context-free subsets provide valuable insights into the study of the word problem and other formal language properties of groups. By characterizing when subsets of a group are recognisably context-free, researchers can better understand the complexity of the word problem in that group. For example, the classification of recognisably context-free conjugacy classes in virtually free groups sheds light on the structure of these groups and the nature of their word problems. Additionally, the connection between recognisably context-free subsets and properties of Cayley graphs helps bridge the gap between geometric representations of groups and their formal language properties. Overall, the study of recognisably context-free subsets enriches the understanding of group theory and its connections to formal languages.