Groups with Context-Free Co-Word Problem and Embeddings into Thompson's Group $V$
Rose Berns-Zieve, Dana Fry, Johnny Gillings, Hannah Hoganson, Heather, Mathews

TL;DR
This paper introduces a class of generalized Thompson groups constructed via cloning systems, proves they have context-free co-word problems, and explores their embeddings and potential as counterexamples to a conjecture relating co-word problems to subgroups of V.
Contribution
It constructs and analyzes the co-word problem of generalized Thompson groups $V_{(G, heta)}$, showing they are co$ ext{CF}$ and extends the class of demonstrative subgroups of V.
Findings
$V_{(G, heta)}$ is co$ ext{CF}$ for any finite group G and homomorphism $ heta$
The co-word problem of $V_{(G, heta)}$ is a cyclic shift of a context-free language
Virtually cyclic groups are now included as demonstrative subgroups of V.
Abstract
Let be a finitely generated group, and let be a finite subset that generates as a monoid. The \emph{word problem of with respect to } consists of all words in the free monoid that are equal to the identity in . The \emph{co-word problem of with respect to } is the complement in of the word problem. We say that a group is \emph{co} if its co-word problem with respect to some (equivalently, any) finite generating set is a context-free language. We describe a generalized Thompson group for each finite group G and homomorphism : . Our group is constructed using the cloning systems introduced by Witzel and Zaremsky. We prove that is co for any homomorphism and finite group G by constructing a pushdown…
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Taxonomy
Topicssemigroups and automata theory · Geometric and Algebraic Topology · Natural Language Processing Techniques
