The Hanna Neumann conjecture for Demushkin groups

Jaikin–Zapirain, Andrei; Shusterman, Mark

doi:10.1016/j.aim.2019.04.013

Cited by 6 publications

(7 citation statements)

References 36 publications

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“…Proof A pro-p version of this result is proved in [16,Proposition 5.1]. The same proof works in our case.…”

Section: Proposition 44supporting

confidence: 57%

The universality of Hughes-free division rings

Jaikin–Zapirain

2021

Sel. Math. New Ser.

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Let $$E*G$$ E ∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to $$E*G$$ E ∗ G -isomorphism, there exists at most one Hughes-free division $$E*G$$ E ∗ G -ring. However, the existence of a Hughes-free division $$E*G$$ E ∗ G -ring $${\mathcal {D}}_{E*G}$$ D E ∗ G for an arbitrary locally indicable group G is still an open question. Nevertheless, $${\mathcal {D}}_{E*G}$$ D E ∗ G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether $${\mathcal {D}}_{E*G}$$ D E ∗ G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists $${\mathcal {D}}_{E[G]}$$ D E [ G ] and it is universal. In Appendix we give a description of $${\mathcal {D}}_{E[G]}$$ D E [ G ] when G is a RFRS group.

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“…Proof A pro-p version of this result is proved in [16,Proposition 5.1]. The same proof works in our case.…”

Section: Proposition 44supporting

confidence: 57%

The universality of Hughes-free division rings

Jaikin–Zapirain

2021

Sel. Math. New Ser.

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“…In the proof of [JS19, Proposition 7.2], it is shown that is an one-relator -module. Thus, we can produce an exact sequence where is a non-trivial cyclic -module.…”

Section: The -Hall Property For Surface Groupsmentioning

confidence: 99%

“…As is a domain, . By [JS19, Corollary 6.2], . Hence, , where is the Euler characteristic of as a -module.…”

Section: The -Hall Property For Surface Groupsmentioning

confidence: 99%

“…It turned out that this new method (with suitable modifications) gave also a new proof of the original strengthened Hanna Neumann conjecture for free groups. In [JS19], Jaikin-Zapirain and Shusterman developed further the pro- part of [Jai17] and showed that the strengthened Hanna Neumann conjecture holds for non-solvable Demushkin pro- groups (the Demushkin pro- groups are the Poincaré duality pro- groups of cohomological dimension 2).…”

Section: Introductionmentioning

confidence: 99%

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The Hanna Neumann conjecture for surface groups

Antolín

Jaikin–Zapirain

2022

Compositio Math.

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The Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957. In 2011, a strengthened version of the conjecture was proved independently by Joel Friedman and by Igor Mineyev. In this paper we show that the strengthened Hanna Neumann conjecture holds not only in free groups but also in non-solvable surface groups. In addition, we show that a retract in a free group and in a surface group is inert. This implies the Dicks–Ventura inertia conjecture for free and surface groups.

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“…Demushkin pro-p groups are Poincare duality pro-p groups of cohomological dimension 2 and can be seen as pro-p analogues of discrete surface groups. Applying the strategy developed in [55], A. Jaikin-Zapirain and M. Shusterman have proved in [58] the strengthened Hanna Neumann conjecture for non-solvable Demuskin pro-p groups. Recall that by Proposition 5.6, if G is an ICC countable group, then Z(R K [G] ) is a subfield of C. Another consequence of Theorem 10.1 is the following result.…”

Section: The Hanna Neumann Conjecturementioning

confidence: 99%