Recent zbMATH articles in MSC 20Jhttps://www.zbmath.org/atom/cc/20J2021-04-16T16:22:00+00:00WerkzeugA characterization of relatively hyperbolic groups via bounded cohomology.https://www.zbmath.org/1456.200522021-04-16T16:22:00+00:00"Franceschini, Federico"https://www.zbmath.org/authors/?q=ai:franceschini.federicoSummary: It was proved by \textit{L. Mineyev} and \textit{A. Yaman} [``Relative hyperbolicity and bounded cohomology'', Preprint] that, if \((\Gamma,\Gamma')\) is a relatively hyperbolic pair, the comparison map
\[
H_b^k(\Gamma,\Gamma';V)\to H^k(\Gamma,\Gamma';V)
\]
is surjective for every \(k\geq2\), and any bounded \(\Gamma\)-module \(V\). By exploiting results of \textit{D. Groves} and \textit{J. F. Manning} [Isr. J. Math. 168, 317--429 (2008; Zbl 1211.20038)], we give another proof of this result. Moreover, we prove the opposite implication under weaker hypotheses than the ones required by Mineyev and Yaman [loc. cit.] .Stability, cohomology vanishing, and nonapproximable groups.https://www.zbmath.org/1456.220022021-04-16T16:22:00+00:00"De Chiffre, Marcus"https://www.zbmath.org/authors/?q=ai:de-chiffre.marcus"Glebsky, Lev"https://www.zbmath.org/authors/?q=ai:glebsky.lev-yu"Lubotzky, Alexander"https://www.zbmath.org/authors/?q=ai:lubotzky.alexander"Thom, Andreas"https://www.zbmath.org/authors/?q=ai:thom.andreas-bertholdThere is substantial current interest in finite-dimensional approximation properties of groups. These generally ask whether every discrete groups can be faithfully represented by approximate homomorphisms into matrix groups, with many possible variations on how exactly this approximation is measured. This paper provides the first negative answer to one of these questions, namely where approximation in the unitary groups is measured by distance in the unnormalized Frobenius norm. (The technical details are given in terms of ultrafilter convergence.)
The paper constructs finitely presented groups for which every approximate homomorphism into a finite-dimensional unitary group must map some (fixed) nontrivial group elements to close to the identity (Theorem 1.1). This follows from two separate results: first, it is shown (Theorem 1.2) that every finitely presented group which has vanishing second cohomology with coefficients in a unitary representation is stable, allowing for the deformation of approximate homomorphisms into actual homomorphisms. This is similar in style to \textit{D. Kazhdan}'s work on stability and cohomology [Isr. J. Math. 43, 315--323 (1982; Zbl 0518.22008)]. Second, it is shown that there are finitely presented groups for which these second cohomology groups vanish, but which are not residually finite. The construction technique here is a \(p\)-adic analogue of Deligne's construction of finite central extensions of arithmetic groups which are not residually finite [\textit{P. Deligne}, C. R. Acad. Sci., Paris, Sér. A 287, 203--208 (1978; Zbl 0416.20042)].
The paper is a pleasure to read and also contains a number of further interesting observations on approximation properties and stability. The problem whether every group is hyperlinear -- which coincides with the question resolved here except for the replacement of the unnormalized Frobenius norm by its normalized version -- remains open.
Reviewer: Tobias Fritz (Innsbruck)The boundary model for the continuous cohomology of \(\mathrm{Isom}^+ (\mathbb H^n)\).https://www.zbmath.org/1456.220042021-04-16T16:22:00+00:00"Pieters, Hester"https://www.zbmath.org/authors/?q=ai:pieters.hesterSummary: We prove that the continuous cohomology of \(\mathrm{Isom}^+ (\mathbb H^n)\) can be measurably realized on the boundary of hyperbolic space. This implies in particular that for \(\mathrm{Isom}^+ (\mathbb H^n)\) the comparison map from continuous bounded cohomology to continuous cohomology is injective in degree 3. We furthermore prove a stability result for the continuous bounded cohomology of \(\mathrm{Isom}(\mathbb H^n)\) and\(\mathrm{Isom}(\mathbb H^n_\mathbb C)\).The Johnson homomorphism and its kernel.https://www.zbmath.org/1456.570162021-04-16T16:22:00+00:00"Putman, Andrew"https://www.zbmath.org/authors/?q=ai:putman.andrewSummary: We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to ``subsurface Torelli groups''. Using this, we extend Johnson's calculation of the rational abelianization of the Torelli group not only to the subsurface Torelli groups, but also to finite-index subgroups of the Torelli group that contain the kernel of the Johnson homomorphism.Cohomology of group theoretic Dehn fillings. I: Cohen-Lyndon type theorems.https://www.zbmath.org/1456.200532021-04-16T16:22:00+00:00"Sun, Bin"https://www.zbmath.org/authors/?q=ai:sun.binDehn surgery is an operation in 3-dimensional topology which consists in modifying a 3-manifold by cutting off a solid torus and then gluing it back in a different way. A famous result in this direction is Thurston's hyperbolic Dehn filling theorem which asserts the existence of hyperbolic structures on a large class of manifolds obtained by Dehn filling. There is also a group theoretic settings for Dehn filling, which had many applications. For instance, the solution of the virtually Haken, by \textit{I. Agol} [Doc. Math. 18, 1045--1087 (2013; Zbl 1286.57019)] uses Dehn fillings of hyperbolic groups.
In the paper under review, the author studies the cohomology of group theoretic Dehn fillings, in particular free product structure of Dehn filling kernels, which he calls the Cohen-Lyndon property. Using this, he describes the structure of relative relation modules of Dehn fillings.
Reviewer: Athanase Papadopoulos (Strasbourg)Limits, standard complexes and \(\mathbf{fr} \)-codes.https://www.zbmath.org/1456.180022021-04-16T16:22:00+00:00"Ivanov, Sergeĭ O."https://www.zbmath.org/authors/?q=ai:ivanov.sergei-o"Mikhailov, Roman V."https://www.zbmath.org/authors/?q=ai:mikhailov.roman"Pavutnitskiy, Fedor Yu."https://www.zbmath.org/authors/?q=ai:pavutnitskiy.fedor-yuLet \(G\) be a group and denote by \(\mathrm{Pres}(G)\) the \textit{category of presentations} of \(G\), whose objects are surjective group morphisms from a free group to \(G\) and morphisms obvious commutative triangles. The article under review deals with nice subfunctors of the functor mapping \(F\twoheadrightarrow G\) (with \(F\) free) to the group ring \(\mathbb{Z}[F]\). One, denoted by \(\mathbf{f}\), is given by the augmentation ideal, and another, denoted by \(\mathbf{g}\), by the kernel of the surjective map \(\mathbb{Z}[F]\twoheadrightarrow\mathbb{Z}[G]\) induced by the presentation: \(\mathbf{f}\) and \(\mathbf{g}\) define functorial (two-sided) ideals of \(\mathbb{Z}[-]\), so that one can build a lot of other ones from them by taking products, intersections or sums. Such ideals are called \(\mathbf{fr}\)-\textit{codes}; for example, \(\mathbf{f}^2\mathbf{r}+(\mathbf{r}^2\cap\mathbf{frf})\) is an \(\mathbf{fr}\)-code. The authors study the problem that they call \textit{translation} of an \(\mathbf{fr}\)-code, that is the description of its higher limits viewed as a functor \(\mathrm{Pres}(G)\to\mathbf{Ab}\).
The article ends with several computational examples, as: \(\lim^i \mathbf{r^2+frf+rf^2}\), which is naturally isomorphic to the group homology \(H_2(G;G_{ab})\) for \(i=1\), to \(G_{ab}^{\otimes 2}\) for \(i=2\) , and to \(0\) for \(i>2\). This is obtained by some general --- elementary but nice -- results in functor cohomology.
Let \(\mathcal{C}\) be a small category and \(X : \mathcal{C}\to\mathbf{Ab}\) a functor. Assume that \(\mathcal{C}\) fulfils both following conditions:
-- for any objects \(c\) and \(d\) of \(\mathcal{C}\), the set \(\mathrm{Hom}(c,d)\) is non-empty;
-- every pair of objects in \(\mathcal{C}\) has a coproduct.
(But one does not require the category to have an initial object, what would imply that all higher limits over \(\mathcal{C}\) are zero. Note that \(\mathrm{Pres}(G)\) always satisfies these conditions.)
It is not hard to see that \(\underset{\mathcal{C}}{\lim}X\) identifies with the equalizer of both canonical arrows \(X(c)\to X(c\sqcup c)\) for any object \(c\) of \(\mathcal{C}\). In Section 2 of the article, the authors generalise as follows this fact: they construct a cosimplicial object in \(\mathcal{C}\) denoted by \(\mathbf{B}(c)\), which is natural in \(c\), such that \(\mathbf{B}(c)^n\) is the coproduct of \((n+1)\) copies of \(c\). They show that the homotopy type of \(\mathbf{B}(c)\) does not depend on \(c\) (\textit{Theorem 1}). Moreover, the cohomotopy of \(X\big(\mathbf{B}(c)\big)\) (meaning the cohomology of the cosimplicial complex given by alternate sum of cofaces, for example) is naturally isomorphic to higher limits of \(X\) (\textit{Theorem 2}).
The article introduces also a notion of (polynomial) degree for a functor \(\mathcal{C}\to\mathbf{Ab}\), which is very much reminiscent of [\textit{S. Eilenberg} and \textit{S. MacLane}, Ann. Math. (2) 60, 49--139 (1954; Zbl 0055.41704)] and, above all, [\textit{T. Pirashvili}, Tr. Tbilis. Mat. Inst. Razmadze 70, 69--91 (1982; Zbl 0521.18015)], and which allows to get vanishing of some higher limits in large cohomological degree (\textit{Proposition 3}). Before applying this to \(\mathbf{fr}\)-codes, the authors give also a Künneth-like formula (\textit{Theorem 4}) in functor cohomology (still in the setting of a source category satisfying the previous assumptions).
Reviewer: Aurelien Djament (Villeneuve d'Ascq)Quasi-automorphisms of the infinite rooted 2-edge-coloured binary tree.https://www.zbmath.org/1456.200592021-04-16T16:22:00+00:00"Nucinkis, Brita E. A."https://www.zbmath.org/authors/?q=ai:nucinkis.brita-e-a"John-Green, Simon St."https://www.zbmath.org/authors/?q=ai:john-green.simon-stSummary: We study the group \(QV\), the self-maps of the infinite 2-edge coloured binary tree which preserve the edge and colour relations at cofinitely many locations. We introduce related groups \(QF\), \(QT\), \(\widetilde{Q}T\), and \(\widetilde{Q}V\), prove that \(QF\), \(\widetilde{Q}T\), and \(\widetilde{Q}V\) are of type \(\mathrm{F}_\infty\), and calculate finite presentations for them. We calculate the normal subgroup structure of all 5 groups, the Bieri-Neumann-Strebel-Renz invariants of \(QF\), and discuss the relationship of all 5 groups with other generalisations of Thompson's groups.Groups acting on trees and the Eilenberg-Ganea problem for families.https://www.zbmath.org/1456.570212021-04-16T16:22:00+00:00"Sánchez Saldaña, Luis Jorge"https://www.zbmath.org/authors/?q=ai:sanchez-saldana.luis-jorgeLet \(G\) be a discrete group and \(\mathcal{F}\) be a family of subgroups of \(G\), by this we mean, a collection of subgroups of \(G\) closed under taking subgroups and conjugation by elements of \(G\). Now let \(E_{\mathcal{F}}G\) be a \textit{universal space} for actions with isotropy in \(\mathcal{F}\). The \(\mathcal{F}\)-\textit{geometric dimension} of \(G\), \(gd_{\mathcal{F}}(G)\), is the minimum nonnegative integer, \(n\), such that there exists an \(n\)-dimensional model for \(E_{\mathcal{F}}G\). On the other hand, let \(H^{*}_{\mathcal{F}}(G;M)\) denote the Bredon cohomology of \(G\) with coefficients in an \(\mathcal{O}_{\mathcal{F}}\)-module \(M\). The \(\mathcal{F}\)-cohomological dimension of \(G\), denoted by \(cd_{\mathcal{F}}(G),\) is the largest nonnegative integer, \(n\), such that \(H^{n}_{\mathcal{F}}(G;M)\) is nontrivial for some \(\mathcal{O}_{\mathcal{F}}\)-module \(M\). It is easy to verify the inequality \(cd_{\mathcal{F}}(G)\leq gd_{\mathcal{F}}(G)\) and it is known
that this is an equality if \(cd_{\mathcal{F}}(G)\geq 3\) and for any family \(\mathcal{F}\). There are some examples, constructed by \textit{N. Brady} et al. [J. Lond. Math. Soc., II. Ser. 64, No. 2, 489--500 (2001; Zbl 1016.20035)] where \(cd_{\mathcal{F}}(G)=2\) and \( gd_{\mathcal{F}}(G)=3\) for the families of finite and virtually cyclic subgroups. Whether this is the case for all families and groups is still an open problem. The purpose of the paper under review is the construction of more examples of groups such that \(cd_{\mathcal{F}}(G)=2\) and \( gd_{\mathcal{F}}(G)=3\) for the families of finite,
virtually abelian of bounded rank and virtually poly-cyclic subgroups of bounded rank. The main technique is the Bass-Serre theory of groups acting on trees.
In order to state the main theorem, we need some definitions. A group \(G\) is called an \(\mathcal{F}\)-\textit{Eilenberg-Ganea} group if \(cd_{\mathcal{F}}(G)=2\) and \( gd_{\mathcal{F}}(G)=3\). Let \(\mathbf{Y}\) be a graph of groups with fundamental group \(G\) (in the Bass-Serre terminology) with associated tree \(T\) and
let \(\mathcal{P}\subset G\) be a collection of subgroups of \(G\). These data are referred to as a splitting of \(G\). The splitting is called
\(\mathcal{P}\)-\textit{acylindrical} if there exists an integer \(k\) such that the stabilizer \(G_{c}\) of a path \(c\subset T\) belongs to \(\mathcal{P}\) for every \(c\) of length \(k\). Furthermore, \(\mathbf{Y}\) is called \(\mathcal{F}\) admissible if the following conditions hold: (1) There exists
a vertex \(P\) such that its vertex group \(Y_{P}\) is a \((Y_{P}\cap\mathcal{F})\)-\textit{Eilenberg-Ganea} group, (2) for all vertices \(P\in \mathbf{Y}\), it follows that \(gd_{\mathcal{F}\cap Y_{P}}(Y_{P})\leq 3\) and \( cd_{\mathcal{F}\cap Y_{P}}(Y_{P})\leq 2\), and (3) for all edges \(y\in \mathbf{Y}\),
we have \( gd_{\mathcal{F}\cap Y_{y}}(Y_{y})\leq 2\) and \( cd_{\mathcal{F}\cap Y_{y}}(Y_{y})\leq 1\). The Main Theorem is as follows:
Theorem. Let \(\mathbf{Y}\) be a graph of groups with fundamental group \(G\) and Bass-Serre tree \(T\). Let \(\mathcal{F}\) be a family of subgroups of \(G\). Let \(\mathcal{F}_{0}\) be the collection of subgroups of \(\mathcal{F}\) that fix a vertex and \(\mathcal{F}_{1}\) those subgroups of \(\mathcal{F}\) that act cocompactly on a geodesic line of \(T\). Assume:
\begin{enumerate}
\item \(\mathbf{Y}\) is \(\mathcal{F}\)-admissible,
\item\(\mathcal{F}=\mathcal{F}_{0}\sqcup \mathcal{F}_{1}\),
\item the splitting of \(G\) is \(\mathcal{P}\)-acylindrical for some collection \(\mathcal{P}\subseteq \mathcal{F}_{0}\) closed under taking subgroups,
\item every \(\mathcal{P}\)-by \(V\) subgroup of \(G\) belongs to \(\mathcal{F}\), where \(V=\mathbb{Z}\) or \(D_{\infty}\),
\item every \(\mathcal{F}_{0}\)-by-\(\mathbb{Z}/2\) subgroup of \(G\) belongs to \(\mathcal{F}_{0}\).
\end{enumerate}
Then \(G\) is an \(\mathcal{F}\)-\textit{Eilenberg-Ganea} group.
In the above, a \(\mathcal{P}\)-by \(Q\) group is a group \(L\) that fits in a short exact sequence \(1\to N\to L\to Q\to 1\) with \(N\in \mathcal{P}\).
Reviewer: Daniel Juan Pineda (Michoacán)