scholarly journals Virtual Retraction Properties in Groups

2019 ◽  
Author(s):  
Ashot Minasyan
Keyword(s):  
Finite Index ◽  
Wreath Products ◽  
Graph Products ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Simple Criterion ◽  
Cyclic Subgroups ◽  
Group A ◽  
Finite Index Subgroup ◽  
Virtually Free Group

Abstract If $G$ is a group, a virtual retract of $G$ is a subgroup, which is a retract of a finite index subgroup. Most of the paper focuses on two group properties: property (LR), that all finitely generated subgroups are virtual retracts; and property (VRC), that all cyclic subgroups are virtual retracts. We study the permanence of these properties under commensurability, amalgams over retracts, graph products, and wreath products. In particular, we show that (VRC) is stable under passing to finite index overgroups, while (LR) is not. The question whether all finitely generated virtually free groups satisfy (LR) motivates the remaining part of the paper, studying virtual free factors of such groups. We give a simple criterion characterizing when a finitely generated subgroup of a virtually free group is a free factor of a finite index subgroup. We apply this criterion to settle a conjecture of Brunner and Burns.

scholarly journals Statistics of subgroups of the modular group

2021 ◽  
pp. 1-61
Author(s):  
Frédérique Bassino ◽  
Cyril Nicaud ◽  
Pascal Weil
Keyword(s):  
Finite Index ◽  
Modular Group ◽  
Large Deviation ◽  
Free Subgroup ◽  
Index Formula ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Finite Index Subgroup ◽  
Free Subgroups ◽  
Finite Index Subgroups

We count the finitely generated subgroups of the modular group [Formula: see text]. More precisely, each such subgroup [Formula: see text] can be represented by its Stallings graph [Formula: see text], we consider the number of vertices of [Formula: see text] to be the size of [Formula: see text] and we count the subgroups of size [Formula: see text]. Since an index [Formula: see text] subgroup has size [Formula: see text], our results generalize the known results on the enumeration of the finite index subgroups of [Formula: see text]. We give asymptotic equivalents for the number of finitely generated subgroups of [Formula: see text], as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size [Formula: see text] subgroup and prove a large deviation statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size [Formula: see text] subgroup (respectively, finite index subgroup, free subgroup) of [Formula: see text].

scholarly journals APPLICATIONS OF p-DEFICIENCY AND p-LARGENESS

2011 ◽  
Vol 21 (04) ◽  
pp. 547-574 ◽  
Author(s):  
J. O. BUTTON ◽  
A. THILLAISUNDARAM
Keyword(s):  
Finite Index ◽  
Coxeter Groups ◽  
Index Subgroup ◽  
Finitely Generated ◽  
P Deficiency ◽  
Finite Presentation ◽  
Finite Index Subgroup ◽  
Finitely Presented

We use Schlage-Puchta's concept of p-deficiency and Lackenby's property of p-largeness to show that a group having a finite presentation with p-deficiency greater than 1 is large, which implies that Schlage-Puchta's infinite finitely generated p-groups are not finitely presented. We also show that for all primes p at least 7, any group having a presentation of p-deficiency greater than 1 is Golod–Shafarevich, and has a finite index subgroup which is Golod–Shafarevich for the remaining primes. We also generalize a result of Grigorchuk on Coxeter groups to odd primes.

scholarly journals Groups with positive rank gradient and their actions

2018 ◽  
Vol 68 (2) ◽  
pp. 353-360
Author(s):  
Mark Shusterman
Keyword(s):  
Finite Index ◽  
Free Groups ◽  
Infinite Index ◽  
Index Subgroup ◽  
Profinite Groups ◽  
Finitely Generated ◽  
Finite Product ◽  
Rank Gradient ◽  
Finite Index Subgroup ◽  
Positive Rank

Abstract We show that given a finitely generated LERF group G with positive rank gradient, and finitely generated subgroups A, B ≤ G of infinite index, one can find a finite index subgroup B0 of B such that [G : 〈A ∪ B0〉] = ∞. This generalizes a theorem of Olshanskii on free groups. We conclude that a finite product of finitely generated subgroups of infinite index does not cover G. We construct a transitive virtually faithful action of G such that the orbits of finitely generated subgroups of infinite index are finite. Some of the results extend to profinite groups with positive rank gradient.

scholarly journals VIRTUAL ALGEBRAIC FIBRATIONS OF KÄHLER GROUPS

2019 ◽  
pp. 1-19
Author(s):  
STEFAN FRIEDL ◽  
STEFANO VIDUSSI
Keyword(s):  
Fundamental Group ◽  
Finite Index ◽  
Surface Groups ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Kähler Surfaces ◽  
Kähler Groups ◽  
Strong Relation ◽  
Finite Index Subgroup ◽  
Finite Index Subgroups

This paper stems from the observation (arising from work of Delzant) that “most” Kähler groups $G$ virtually algebraically fiber, that is, admit a finite index subgroup that maps onto $\mathbb{Z}$ with finitely generated kernel. For the remaining ones, the Albanese dimension of all finite index subgroups is at most one, that is, they have virtual Albanese dimension $va(G)\leqslant 1$ . We show that the existence of algebraic fibrations has implications in the study of coherence and higher BNSR invariants of the fundamental group of aspherical Kähler surfaces. The class of Kähler groups with $va(G)=1$ includes virtual surface groups. Further examples exist; nonetheless, they exhibit a strong relation with surface groups. In fact, we show that the Green–Lazarsfeld sets of groups with $va(G)=1$ (virtually) coincide with those of surface groups, and furthermore that the only virtually RFRS groups with $va(G)=1$ are virtually surface groups.

scholarly journals From local to global conjugacy of subgroups of relatively hyperbolic groups

2017 ◽  
Vol 27 (03) ◽  
pp. 299-314
Author(s):  
Oleg Bogopolski ◽  
Kai-Uwe Bux
Keyword(s):  
Finite Index ◽  
Minimal Length ◽  
Hyperbolic Groups ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Limit Group ◽  
Relatively Hyperbolic Groups ◽  
Finitely Generated Group ◽  
Finite Index Subgroup ◽  
Relatively Hyperbolic

Suppose that a finitely generated group [Formula: see text] is hyperbolic relative to a collection of subgroups [Formula: see text]. Let [Formula: see text] be subgroups of [Formula: see text] such that [Formula: see text] is relatively quasiconvex with respect to [Formula: see text] and [Formula: see text] is not parabolic. Suppose that [Formula: see text] is elementwise conjugate into [Formula: see text]. Then there exists a finite index subgroup of [Formula: see text] which is conjugate into [Formula: see text]. The minimal length of the conjugator can be estimated. In the case, where [Formula: see text] is a limit group, it is sufficient to assume only that [Formula: see text] is a finitely generated and [Formula: see text] is an arbitrary subgroup of [Formula: see text].

Greenberg's Theorem for Quasiconvex Subgroups of Word Hyperbolic Groups

1996 ◽  
Vol 48 (6) ◽  
pp. 1224-1244 ◽  
Author(s):  
Ilya Kapovich ◽  
Hamish Short
Keyword(s):  
Finite Index ◽  
Free Groups ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Word Hyperbolic Group ◽  
Quasiconvex Subgroups ◽  
Word Hyperbolic Groups ◽  
Finite Index Subgroup

AbstractAnalogues of a theorem of Greenberg about finitely generated subgroups of free groups are proved for quasiconvex subgroups of word hyperbolic groups. It is shown that a quasiconvex subgroup of a word hyperbolic group is a finite index subgroup of only finitely many other subgroups.

scholarly journals Contractible, hyperbolic but non-CAT(0) complexes

2020 ◽  
Vol 30 (5) ◽  
pp. 1439-1463
Author(s):  
Richard C. H. Webb
Keyword(s):  
Isoperimetric Inequality ◽  
Finite Index ◽  
Mapping Class Group ◽  
Class Group ◽  
The Other ◽  
Mapping Class ◽  
Index Subgroup ◽  
Analogous Statement ◽  
Almost All ◽  
Finite Index Subgroup

AbstractWe prove that almost all arc complexes do not admit a CAT(0) metric with finitely many shapes, in particular any finite-index subgroup of the mapping class group does not preserve such a metric on the arc complex. We also show the analogous statement for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups. The obstruction is combinatorial. These complexes are all hyperbolic and contractible but despite this we show that they satisfy no combinatorial isoperimetric inequality: for any n there is a loop of length 4 that only bounds discs consisting of at least n triangles. On the other hand we show that the curve complexes satisfy a linear combinatorial isoperimetric inequality, which answers a question of Andrew Putman.

scholarly journals Groups of p-deficiency one

2013 ◽  
Vol 156 (1) ◽  
pp. 115-121
Author(s):  
ANITHA THILLAISUNDARAM
Keyword(s):  
Finite Index ◽  
Index Subgroup ◽  
P Deficiency ◽  
Finitely Presented Groups ◽  
Finite Presentation ◽  
Finite Index Subgroup ◽  
Finitely Presented

AbstractIn a previous paper, Button and Thillaisundaram proved that all finitely presented groups of p-deficiency greater than one are p-large. Here we prove that groups with a finite presentation of p-deficiency one possess a finite index subgroup that surjects onto the integers. This implies that these groups do not have Kazhdan's property (T). Additionally, we show that the aforementioned result of Button and Thillaisundaram implies a result of Lackenby.

scholarly journals An Abramov formula for stationary spaces of discrete groups

2013 ◽  
Vol 34 (3) ◽  
pp. 837-853 ◽  
Author(s):  
YAIR HARTMAN ◽  
YURI LIMA ◽  
OMER TAMUZ
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Finite Index ◽  
Discrete Group ◽  
Return Time ◽  
Discrete Groups ◽  
Poisson Boundary ◽  
Index Subgroup ◽  
Expected Return ◽  
Finite Index Subgroup

AbstractLet $(G, \mu )$ be a discrete group equipped with a generating probability measure, and let $\Gamma $ be a finite index subgroup of $G$. A $\mu $-random walk on $G$, starting from the identity, returns to $\Gamma $ with probability one. Let $\theta $ be the hitting measure, or the distribution of the position in which the random walk first hits $\Gamma $. We prove that the Furstenberg entropy of a $(G, \mu )$-stationary space, with respect to the action of $(\Gamma , \theta )$, is equal to the Furstenberg entropy with respect to the action of $(G, \mu )$, times the index of $\Gamma $ in $G$. The index is shown to be equal to the expected return time to $\Gamma $. As a corollary, when applied to the Furstenberg–Poisson boundary of $(G, \mu )$, we prove that the random walk entropy of $(\Gamma , \theta )$ is equal to the random walk entropy of $(G, \mu )$, times the index of $\Gamma $ in $G$.

ON KAZHDAN CONSTANTS OF FINITE INDEX SUBGROUPS IN SLn(ℤ)

2012 ◽  
Vol 22 (03) ◽  
pp. 1250026
Author(s):  
UZY HADAD
Keyword(s):  
Finite Index ◽  
Congruence Subgroup ◽  
Infinite Family ◽  
Principal Congruence ◽  
The Other ◽  
Index Subgroup ◽  
Generating Set ◽  
Principal Congruence Subgroup ◽  
Finite Index Subgroup ◽  
Finite Index Subgroups

We prove that for any finite index subgroup Γ in SL n(ℤ), there exists k = k(n) ∈ ℕ, ϵ = ϵ(Γ) > 0, and an infinite family of finite index subgroups in Γ with a Kazhdan constant greater than ϵ with respect to a generating set of order k. On the other hand, we prove that for any finite index subgroup Γ of SL n(ℤ), and for any ϵ > 0 and k ∈ ℕ, there exists a finite index subgroup Γ′ ≤ Γ such that the Kazhdan constant of any finite index subgroup in Γ′ is less than ϵ, with respect to any generating set of order k. In addition, we prove that the Kazhdan constant of the principal congruence subgroup Γn(m), with respect to a generating set consisting of elementary matrices (and their conjugates), is greater than [Formula: see text], where c > 0 depends only on n. For a fixed n, this bound is asymptotically best possible.

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