Recent zbMATH articles in MSC 20J06https://www.zbmath.org/atom/cc/20J062021-04-16T16:22:00+00:00WerkzeugStability, 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.A 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.] .