Fixed subgroups in free groups: a survey

Capell, Enric Ventura

doi:10.1090/conm/296/05077

Cited by 32 publications

(26 citation statements)

References 53 publications

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“…Recall that H is compressed if rk(H) ≤ rk(K) for every K ≤ F containing H (see Section 3.4), and say that H is inert if rk(H ∩K) ≤ rk(K) for every K ≤ F . Both these properties were introduced by Dicks and Ventura [5] in the context of the study of subgroups of free groups that are fixed by sets of endomorphisms or automorphisms (see also [21]). …”

Section: Proposition 46 Let H ≤ F (A) Be An Extension Of Finitely Gementioning

confidence: 99%

Algebraic Extensions in Free Groups

Miasnikov

Ventura

Weil

2007

Geometric Group Theory

106

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The aim of this paper is to unify the points of view of three recent and independent papers (Ventura 1997, Margolis, Sapir and Weil 2001and Kapovich and Miasnikov 2002, where similar modern versions of a 1951 theorem of Takahasi were given. We develop a theory of algebraic extensions for free groups, highlighting the analogies and differences with respect to the corresponding classical field-theoretic notions, and we discuss in detail the notion of algebraic closure. We apply that theory to the study and the computation of certain algebraic properties of subgroups (e.g. being malnormal, pure, inert or compressed, being closed in certain profinite topologies) and the corresponding closure operators. We also analyze the closure of a subgroup under the addition of solutions of certain sets of equations. *

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Section: Proposition 46 Let H ≤ F (A) Be An Extension Of Finitely Gementioning

confidence: 99%

Algebraic Extensions in Free Groups

Miasnikov

Ventura

Weil

2007

Geometric Group Theory

106

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“…Goldstein and Turner extended it to monomorphisms of free groups [11], and later to arbitrary endomorphisms [12]. Collins and Turner extended it to automorphisms of free products of freely indecomposable groups [3] (see the survey by Ventura [22]). With respect to automorphisms, the widest generalization is to hyperbolic groups and is due to Paulin [15].…”

Section: Introductionmentioning

confidence: 99%

Fixed points of endomorphisms of virtually free groups

Silva

2013

Pacific J. Math.

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A fixed point theorem is proved for inverse transducers, leading to an automatatheoretic proof of the fixed point subgroup of an endomorphism of a finitely generated virtually free group being finitely generated. If the endomorphism is uniformly continuous for the hyperbolic metric, it is proved that the set of regular fixed points in the hyperbolic boundary has finitely many orbits under the action of the finite fixed points. In the automorphism case, it is shown that these regular fixed points are either exponentially stable attractors or exponentially stable repellers.

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“…(iii) In [5], the previous result was generalised to say that every 1-mono-ÿxed subgroup H of F is F-inert, i.e. r(H ∩ K) 6 r(K) for every K 6 F. Inertia for 1-endo-ÿxed subgroups is an open problem (see [1,17,13] for related results).…”

Section: Letmentioning

confidence: 99%

A description of auto-fixed subgroups in a free group

Martino

Ventura

2004

Topology

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Let F be a finitely generated free group. By using Bestvina-Handel theory, as well\ud as some further improvements, the eigengroups of a given automorphism of F (and\ud its fixed subgroup among them) are globally analyzed and described. In particular,\ud an explicit description of all subgroups of F which occur as the fixed subgroup of\ud some automorphism is given.Peer ReviewedPostprint (author’s final draft

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Fixed subgroups in free groups: a survey

Cited by 32 publications

References 53 publications

Algebraic Extensions in Free Groups

Algebraic Extensions in Free Groups

Fixed points of endomorphisms of virtually free groups

A description of auto-fixed subgroups in a free group

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