1,353 research outputs found

    Conjugacy in normal subgroups of hyperbolic groups

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    Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G. We establish criteria for N to have solvable conjugacy problem and be conjugacy separable in terms of the corresponding properties of G/N. We show that the hyperbolic group from F. Haglund's and D. Wise's version of Rips's construction is hereditarily conjugacy separable. We then use this construction to produce first examples of finitely generated and finitely presented conjugacy separable groups that contain non-(conjugacy separable) subgroups of finite index.Comment: Version 3: 18 pages; corrected a problem with justification of Corollary 8.

    Subgroups of direct products of limit groups over Droms RAAGs

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    A result of Bridson, Howie, Miller and Short states that if SS is a subgroup of type FPn(Q)FP_{n}(\mathbb{Q}) of the direct product of nn limit groups over free groups, then SS is virtually the direct product of limit groups over free groups. Furthermore, they characterise finitely presented residually free groups. In this paper these results are generalised to limit groups over Droms right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with the property that all of their finitely generated subgroups are again RAAGs. In addition, we show that the generalised conjugacy problem is solvable for finitely presented groups that are residually a Droms RAAG and that the membership problem is decidable for their finitely presented subgroups

    Subgroups of direct products of limit groups over Droms RAAGs

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    A result of Bridson, Howie, Miller and Short states that if S is a subgroup of type FPn(Q) of the direct product of n limit groups over free groups, then S is virtually the direct product of limit groups over free groups. Furthermore, they characterise finitely presented residually free groups. In this paper these results are generalised to limit groups over Droms right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with the property that all of their finitely generated subgroups are again RAAGs. In addition, we show that the generalised conjugacy problem is solvable for finitely presented groups that are residually a Droms RAAG, and that their finitely presentable subgroups are separable

    Orbit decidability and the conjugacy problem for some extensions of groups

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    Given a short exact sequence of groups with certain conditions, 1β†’Fβ†’Gβ†’Hβ†’11\to F\to G\to H\to 1, we prove that GG has solvable conjugacy problem if and only if the corresponding action subgroup Aβ©½Aut(F)A\leqslant Aut(F) is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form Z2β‹ŠFm\mathbb{Z}^2\rtimes F_m, F2β‹ŠFmF_2\rtimes F_m, Fnβ‹ŠZF_n \rtimes \mathbb{Z}, and Znβ‹ŠAFm\mathbb{Z}^n \rtimes_A F_m with virtually solvable action group Aβ©½GLn(Z)A\leqslant GL_n(\mathbb{Z}). Also, we give an easy way of constructing groups of the form Z4β‹ŠFn\mathbb{Z}^4\rtimes F_n and F3β‹ŠFnF_3\rtimes F_n with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in Aut(F2)Aut(F_2) is given

    The membership problem for 3-manifold groups is solvable

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    We show that the Membership Problem for finitely generated subgroups of 3-manifold groups is solvable.Comment: 19 page

    Decision problems and profinite completions of groups

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    We consider pairs of finitely presented, residually finite groups P\hookrightarrow\G for which the induced map of profinite completions \hat P\to \hat\G is an isomorphism. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not PP is isomorphic to \G. We construct pairs for which the conjugacy problem in \G can be solved in quadratic time but the conjugacy problem in PP is unsolvable. Let J\mathcal J be the class of super-perfect groups that have a compact classifying space and no proper subgroups of finite index. We prove that there does not exist an algorithm that, given a finite presentation of a group \G and a guarantee that \G\in\mathcal J, can determine whether or not \G\cong\{1\}. We construct a finitely presented acyclic group \H and an integer kk such that there is no algorithm that can determine which kk-generator subgroups of \H are perfect

    Structure and finiteness properties of subdirect products of groups

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    We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands. This leads to a characterization of the finitely presented subgroups of the direct product of 3 free or surface groups, and to a solution to the conjugacy problem for arbitrary finitely presented subgroups of direct products of surface groups. We obtain a formula for the first homology of a subdirect product of two free groups and use it to show there is no algorithm to determine the first homology of a finitely generated subgroup.Comment: 29 pages, no figure

    The Geometry of the Conjugacy Problem in Wreath Products and Free Solvable Groups

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    We describe an effective version of the conjugacy problem and study it for wreath products and free solvable groups. The problem involves estimating the length of short conjugators between two elements of the group, a notion which leads to the definition of the conjugacy length function. We show that for free solvable groups the conjugacy length function is at most cubic. For wreath products the behaviour depends on the conjugacy length function of the two groups involved, as well as subgroup distortion within the quotient group.Comment: 24 pages, 4 figures. This was formed from the splitting of arXiv:1202.5343, titled "On the Magnus Embedding and the Conjugacy Length Function of Wreath Products and Free Solvable Groups," into two papers. The contents of this paper remain largely unchange
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