Hyperbolic surface subgroups of one-ended doubles of free groups

Kim, Sang-Hyun; Oum, Sang‐il

doi:10.1112/jtopol/jtu004

Cited by 7 publications

(6 citation statements)

References 22 publications

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“…(3) Fundamental groups of hyperbolic 3-manifolds (Kahn-Markovic [19]); (4) Certain doubles of free groups (Gordon-Wilton, Kim-Wilton, Kim-Oum [15,21,20]);…”

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confidence: 99%

“…(1) A group G with b 2 > 0 obtained by doubling a free group F along a finite collection of finitely generated subgroups F i ; (2) A group G obtained as an HNN extension F * φ where F is a free group of fixed rank and φ is a random endomorphism; (3) "Sapir's group" C = F * φ for F = a, b and φ : a → ab, b → ba. The sense in which this constitutes a significant advance over the results and methods in [4,21,20] is that the edge groups are free groups of arbitrary rank, whereas in the cited papers the edge groups were required to be cyclic.…”

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confidence: 99%

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Surface subgroups from linear programming

Calegari¹,

Walker²

2015

Duke Math. J.

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Abstract. We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive b 2 obtained by doubling free groups along collections of subgroups, and groups obtained by "random" ascending HNN extensions of free groups. A special case is the HNN extension associated to the endomorphism of a rank 2 free group sending a to ab and b to ba; this example (and the random examples) answer in the negative well-known questions of Sapir. We further show that the unit ball in the Gromov norm (in dimension 2) of a double of a free group along a collection of subgroups is a finite-sided rational polyhedron, and that every rational class is virtually represented by an extremal surface subgroup. These results are obtained by a mixture of combinatorial, geometric, and linear programming techniques.

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“…(3) Fundamental groups of hyperbolic 3-manifolds (Kahn-Markovic [19]); (4) Certain doubles of free groups (Gordon-Wilton, Kim-Wilton, Kim-Oum [15,21,20]);…”

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confidence: 99%

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confidence: 99%

Surface subgroups from linear programming

Calegari¹,

Walker²

2015

Duke Math. J.

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“…Calegari [4] showed that (under certain homological assumptions) hyperbolic groups obtained as graphs of free groups amalgamated over cyclic subgroups contain surface subgroups. Kim and Oum [15] showed the same is true if G is a one-ended double of a free group (this work was preceded by the paper of Gordon and Wilton [10]).…”

Section: Introductionmentioning

confidence: 71%

“…Consider the restriction of φ U to some D H . Then (19) follows from (18) and from (15) in Proposition 3.6. Hence, we have proved that φ U conjugates the action of µ(G) from S 2 to ∂U .…”

Section: 4mentioning

confidence: 94%

Criterion for Cannon’s Conjecture

Marković

2013

Geom. Funct. Anal.

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The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon's Conjecture: A hyperbolic group G (that acts effectively on its boundary) whose boundary is homeomorphic to the 2-sphere is isomorphic to a Kleinian group if and only if every two points in the boundary of G are separated by a quasi-convex surface subgroup. Thus, the Cannon's conjecture is reduced to showing that such a group contains "enough" quasi-convex surface subgroups.When ∂G ≈ S 2 , quasi-convex surface subgroups are examples of codimension 1 subgroups of G. If every two points in ∂G can be separated by a quasi-convex surface subgroup we say that G contains enough quasi-convex CRITERION FOR CANNON'S CONJECTURE 3 surface subgroups. The main result of this paper is the following criterion for the Cannon's Conjecture:Theorem 1.1 (Criterion for Cannon's Conjecture). Let G be a hyperbolic group (that acts effectively on its boundary) with ∂G ≈ S 2 . Suppose that G contains enough quasi-convex surface subgroups. Then G is isomorphic to a Kleinian group.Kahn and Markovic [12] showed that the fundamental group of a closed hyperbolic 3-manifold contains enough quasi-convex surface subgroups and thus by the above theorem we have the following equivalence stated in the abstract above:A hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group if and only if it contains enough quasi-convex surface subgroups. 1.2. Cubulation and separability in hyperbolic groups. The sizable part of the argument behind the proof of the Criterion for Cannon's conjecture relies on the following theorem recently proved by Agol (see Theorem 1.1 and Corollary 1.2 in [1]): Theorem 1.2 (Agol). Let G be a cubulated hyperbolic group (a group is cubulated if it is acting properly and co-compactly on a CAT(0) cube complex). Then quasi-convex subgroups of G are separable.Remark. This theorem is stated as Corollary 1.2 in [1]. If a hyperbolic groups G is cubulated then it is proved that G virtually embeds into a Right Angled Artin Group and thus G has a finite index subgroup whose quasiconvex subgroups are separable. On the other hand, the notion of quasiconvex subgroups being separable is invariant under passing onto subgroups or onto finite index supergroups (see Lemma 2.2.2. in [16]) and we conclude that quasi-convex subgroups of G are separable.Recall that a subgroup H < G is separable if there is a sequence G n < G of finite index subgroups of G such thatAlso, for a subgroup H < G and g ∈ G we will use the standard abbreviation H g = g −1 Hg. Another important ingredient we need is the following theorem of Bergeron-Wise (see the statement and the proof of Theorem 1.4 in [3]) that builds on the work of Sageev [18]: Theorem 1.3 (Bergeron-Wise). Let G be a hyperbolic group that contains enough quasi-convex,...

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“…Using the hierarchical structure of limit groups described above, one sees that the crucial case to consider is one where Γ is the fundamental group of a graph of free groups with cyclic edge groups. There has been much recent work on the existence of surface subgroups in such groups; see [14], [22], [30], and [29] for example. Remark 4.19.…”

Section: Residually Free Groupsmentioning

confidence: 99%

Determining Fuchsian groups by their finite quotients

2016

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Let C(Γ) be the set of isomorphism classes of the finite groups that are quotients (homomorphic images) of Γ. We investigate the extent to which C(Γ) determines Γ when Γ is a group of geometric interest. If Γ1 is a lattice in PSL(2, R) and Γ2 is a lattice in any connected Lie group, then C(Γ1) = C(Γ2) implies that Γ1 ∼ = Γ2. If F is a free group and Γ is a right-angled Artin group or a residually free group (with one extra condition), then C(F ) = C(Γ) implies that F ∼ = Γ. If Γ1 < PSL(2, C) and Γ2 < G are non-uniform arithmetic lattices, where G is a semi-simple Lie group with trivial centre and no compact factors, then C(Γ1) = C(Γ2) implies that G ∼ = PSL(2, C) and that Γ2 belongs to one of finitely many commensurability classes.These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two nonisomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them.In this section we recall some background on profinite completions and the theory of profinite groups; see [48], [50] and [52] for more details.

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Hyperbolic surface subgroups of one-ended doubles of free groups

Cited by 7 publications

References 22 publications

Surface subgroups from linear programming

Surface subgroups from linear programming

Criterion for Cannon’s Conjecture

Determining Fuchsian groups by their finite quotients

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