A list of applications of Stallings automata

Delgado, Jordi; Ventura, Enric

doi:10.48550/arxiv.2109.01268

Cited by 2 publications

(2 citation statements)

References 68 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Can the intersection of two f.g. ideals be described and is there a bound on its rank as in the Hanna Neumann conjecture? See the recent survey [DV21] giving a list of results about free groups and their subgroups using Stalling core graphs.…”

Section: Appendix a The Limit Distribution Of Fixmentioning

confidence: 99%

Word Measures on $GL_N(q)$ and Free Group Algebras

Ernst-West¹,

Puder²,

Seidel³

2021

Preprint

View full text Add to dashboard Cite

Fix a finite field K of order q and a word w in a free group F on r generators. A wrandom element in GL N (K) is obtained by sampling r independent uniformly random elements g 1 , . . . , g r ∈ GL N (K) and evaluating w (g 1 , . . . , g r ). Consider E w [fix], the average number of vectors in K N fixed by a w-random element. We show that E w [fix] is a rational function in q N . Moreover, if w = u d with u a non-power, then the limit lim N →∞ E w [fix] depends only on d and not on u. These two phenomena generalize to all stable characters of the groups {GL N (K)} N .A main feature of this work is the connection we establish between word measures on GL N (K) and the free group algebra K-module with a well-defined rank. We show that for, where C is the number of rank-2 right ideals I ≤ K [F] which contain w − 1 but not as a basis element. We describe a full conjectural picture generalizing this result.In the process, we prove several new results about free group algebras. For example, we show that if T is any finite subtree of the Cayley graph of F, and I ≤ K [F] is a right ideal with a generating set supported on T , then I admits a basis supported on T . We also prove an analogue of Kaplansky's unit conjecture for certain K [F]-modules.

show abstract

Section: Appendix a The Limit Distribution Of Fixmentioning

confidence: 99%

Word Measures on $GL_N(q)$ and Free Group Algebras

Ernst-West¹,

Puder²,

Seidel³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Besides being a very elegant theory, this approach displayed great benefits from an algorithmic viewpoint. For a list of applications of Stallings automata, see [4].…”

Section: Introductionmentioning

confidence: 99%

On the transition monoid of the Stallings automaton of a subgroup of a free group

Guimarães¹

2021

Preprint

View full text Add to dashboard Cite

Birget, Margolis, Meakin and Weil proved that a finitely generated subgroup K of a free group is pure if and only if the transition monoid M (K) of its Stallings automaton is aperiodic. In this paper, we establish further connections between algebraic properties of K and algebraic properties of M (K). We mainly focus on the cases where M (K) belongs to the pseudovariety H of finite monoids all of whose subgroups lie in a given pseudovariety H of finite groups. We also discuss normal, malnormal and cyclonormal subgroups of F A using the transition monoid of the corresponding Stallings automaton.

show abstract

A list of applications of Stallings automata

Cited by 2 publications

References 68 publications

Word Measures on $GL_N(q)$ and Free Group Algebras

Word Measures on $GL_N(q)$ and Free Group Algebras

On the transition monoid of the Stallings automaton of a subgroup of a free group

Contact Info

Product

Resources

About