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Register a new userInduced quasi-cocycles on groups with hyperbolically embedded subgroups
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Let G be a group, H a hyperbolically embedded subgroup of G, V a normed G-module, U an H-invariant submodule of V. We propose a general construction which allows to extend 1-quasi-cocycles on H with values in U to 1-quasi-cocycles on G with values in V. As an application, we show that every group G with a non-degenerate hyperbolically embedded subgroup has dim H^2_b (G, l^p(G))=infty for pin [1, infty). This covers many previously known results in a uniform way. Applying our extension to quasimorphisms and using Bavard duality, we also show that hyperbolically embedded subgroups are undistorted with respect to the stable commutator length.
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Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${rm Out}(F_n)$, dots) embeds via the natural restriction maps into the inverse limit of the second bounded cohomologies of its virtually free subgroups, and in fact even into the inverse limit of the second bounded cohomologies of its hyperbolically embedded virtually free subgroups. This result is new and non-trivial even in the case where $G$ is a (non-free) hyperbolic group. The corresponding statement fails in general for the third bounded cohomology, even for surface groups.
We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G,X,H) consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H which is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of Out(Fn), and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the study of intersection and combination properties of A/QI triples. Under the additional hypothesis that G is finitely generated, we use a method of Sisto to show that H is stable. We apply theorems of Kapovich--Rafi and Dowdall--Taylor to analyze the Gromov boundary of an associated cone-off. We close with some examples and questions.
This article extends the works of Gonc{c}alves, Guaschi, Ocampo [GGO] and Marin [MAR2] on finite subgroups of the quotients of generalized braid groups by the derived subgroup of their pure braid group. We get explicit criteria for subgroups of the (complex) reflection group to lift to subgroups of this quotient. In the specific case of the classical braid group, this enables us to describe all its finite subgroups : we show that every odd-order finite group can be embedded in it, when the number of strands goes to infinity. We also determine a complete list of the irreducible reflection groups for which this quotient is a Bieberbach group.
For each natural number $d$ we construct a $3$-generated group $H_d$, which is a subdirect product of free groups, such that the cohomological dimension of $H_d$ is $d$. Given a group $F$ and a normal subgroup $N lhd F$ we prove that any right angled Artin group containing the special HNN-extension of $F$ with respect to $N$ must also contain $F/N$. We apply this to construct, for every $d in mathbb{N}$, a $4$-generated group $G_d$, embeddable into a right angled Artin group, such that the cohomological dimension of $G_d$ is $2$ but the cohomological dimension of any right angled Artin group, containing $G_d$, is at least $d$. These examples are used to show the non-existence of certain universal right angled Artin groups. We also investigate finitely presented subgroups of direct products of limit groups. In particular we show that for every $nin mathbb{N}$ there exists $delta(n) in mathbb{N}$ such that any $n$-generated finitely presented subgroup of a direct product of finitely many free groups embeds into the $delta(n)$-th direct power of the free group of rank $2$. As another corollary we derive that any $n$-generated finitely presented residually free group embeds into the direct product of at most $delta(n)$ limit groups.
Let $G$ be a finite group and $sigma ={sigma_{i} | iin I}$ some partition of the set of all primes $Bbb{P}$, that is, $sigma ={sigma_{i} | iin I }$, where $Bbb{P}=bigcup_{iin I} sigma_{i}$ and $sigma_{i}cap sigma_{j}= emptyset $ for all $i e j$. We say that $G$ is $sigma$-primary if $G$ is a $sigma _{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: ${sigma}$-subnormal in $G$ if there is a subgroup chain $A=A_{0} leq A_{1} leq cdots leq A_{n}=G$ such that either $A_{i-1}trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $sigma$-primary for all $i=1, ldots, n$, modular in $G$ if the following conditions hold: (i) $langle X, A cap Z rangle=langle X, A rangle cap Z$ for all $X leq G, Z leq G$ such that $X leq Z$, and (ii) $langle A, Y cap Z rangle=langle A, Y rangle cap Z$ for all $Y leq G, Z leq G$ such that $A leq Z$. In this paper, a subgroup $A$ of $G$ is called $sigma$-quasinormal in $G$ if $L$ is modular and ${sigma}$-subnormal in $G$. We study $sigma$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $sigma$-quasinormal in $G$, then for every chief factor $H/K$ of $G$ between $H^{G}$ and $H_{G}$ the semidirect product $(H/K)rtimes (G/C_{G}(H/K))$ is $sigma$-primary.
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