Recent zbMATH articles in MSC 57Shttps://zbmath.org/atom/cc/57S2021-05-28T16:06:00+00:00WerkzeugOn upper and lower bounds for finite group-actions on bounded surfaces, handlebodies, closed handles and finite graphs.https://zbmath.org/1459.570282021-05-28T16:06:00+00:00"Zimmermann, Bruno P."https://zbmath.org/authors/?q=ai:zimmermann.bruno-pThe author provides a good survey of known results and proves new results about the maximal order of finite group actions
on spaces such as (i) surfaces with non-empty boundary and \(3\)-dimensional handlebodies (ii) connected sums of copies of \(S^1{\times}S^2\) (iii) higher dimensional handlebodies, and (iv) finite graphs.
A typical new result is the following: Let \(m(g)\) be the maximal order of a free, orientation-preserving finite group action on a connected sum of \(g>1\) copies of \(S^1{\times}S^2\). Then (a) \(2(g+1)\leq m(g)\leq 6(g-1)\). (b) If \(g\) is odd, then \(m(g)=4(g-1)\) or \(6(g-1)\) and both cases occur for infinitely many \(g\). (c) The possible values for \(m(g)\) are \(\frac{2n}{n-2}(g-1)\) for integers \(n\geq 3\), and infinitely many values of \(n\) occur, resp. do not occur.
The same results hold for finite orientation-preserving group actions on surfaces with boundary of algebraic rank \(g\) (\(=\) the rank of the free fundamental group).
A similar result is proved for finite group actions on finite graphs.
Reviewer: Wolfgang Heil (Tallahassee)The \(RO(C_2)\)-graded cohomology of \(C_2\)-surfaces in \(\underline{\mathbb{Z} / 2}\)-coefficients.https://zbmath.org/1459.570272021-05-28T16:06:00+00:00"Hazel, Christy"https://zbmath.org/authors/?q=ai:hazel.christyThe author computes the \(RO(C_2)\)-graded Bredon cohomology of all closed \(C_2\)-surfaces with \(\mathbb Z_2\)-coefficients as modules over the cohomology of a point. The cohomology depends on three numerical invariants if the action is nonfree and two if the action is free. Among the invariants is the number of fixed points and fixed circles. The work is based on the classification of involutions on surfaces by [\textit{D. Dugger}, J. Homotopy Relat. Struct. 14, No. 4, 919--992 (2019; Zbl 1431.57026)].
Reviewer: Karl Heinz Dovermann (Honolulu)Three perfect mapping class groups.https://zbmath.org/1459.570222021-05-28T16:06:00+00:00"Vlamis, Nicholas G."https://zbmath.org/authors/?q=ai:vlamis.nicholas-gSummary: We prove that the mapping class group of a surface obtained from removing a Cantor set from either the 2-sphere, the plane, or the interior of the closed 2-disk has no proper countable-index normal subgroups. The proof is an application of the automatic continuity of these groups, which was established by \textit{K. Mann} [``Automatic continuity for homeomorphism groups of noncompact manifolds'', Preprint, \url{arXiv:2003.01173}]. As corollaries, we see that these groups do not contain any proper finite-index subgroups and that each of these groups has trivial abelianization.Finite groups of rank two which do not involve \(\mathrm{Qd}(p)\).https://zbmath.org/1459.200182021-05-28T16:06:00+00:00"Kızmaz, Muhammet Yasir"https://zbmath.org/authors/?q=ai:kizmaz.muhammet-yasir"Yalçın, Ergün"https://zbmath.org/authors/?q=ai:yalcin.ergunA \(p\)-group is said to be of rank \(k\) if the largest possible order of an elementary abelian subgroup of the group is \(p^k\). Further, the \(p\)-rank of a finite group \(G\) is \(k\) if a Sylow \(p\)-subgroup of \(G\) is of rank \(k\). The invariant \(p\)-rank of a finite group is conjecturally related to the minimum product of spheres on which the finite group can act freely.
Let \(Qd(p) := (\mathbb{Z}/p \times \mathbb{Z}/p) \rtimes SL(2, p)\), where \(\mathrm{SL}(2, p)\) acts on \(\mathbb{Z}/p \times \mathbb{Z}/p\) in the usual manner with \(\mathbb{Z}/p \times \mathbb{Z}/p\) viewed as a two-dimensional vector space over the field of \(p\)-elements. A finite group \(G\) is said to involve \(Qd(p)\) if there exist subgroups \(K \triangleleft H \le G\) such that \(H/K \cong Qd(p)\). Further, \(G\) is said to \(p'\)-involve \(Qd(p)\) if there exist \(K \triangleleft H \le G\) such that \(K\) has order coprime to \(p\) and \(H/K \cong Qd(p)\). If a group \(p'\)-involves \(Qd(p)\), then obviously it involves \(Qd(p)\), but the converse does not hold in general.
One of the main results of the paper is that these two conditions are equivalent for finite groups with \(p\)-rank equal to two, where \(p > 3\). The result is no longer true for \(p =2, 3\). The main ingredient in the proof is the classification of \(p\)-groups of rank two. The result allows the authors to use a version of Glauberman's ZJ-theorem to give a more direct construction of finite group actions on mod-\(p\) homotopy spheres.
Reviewer: Mahender Singh (Sahibzada Ajit Singh Nagar)