Group actions on contractible 222-complexes II (original) (raw)

Partial Actions of Groups on Cell Complexes

Monatshefte f�r Mathematik, 2003

In this paper, we study partial group actions on 2-complexes. Our results include a characterization, in terms of generating sets, of when a partial group action on a connected 2-complex has a connected globalization. Using this result, we give a short combinatorial proof that a group acting without fixed points on a connected 2-complex, with finite quotient, is finitely generated. This result is then generalized to characterize finitely generated groups as precisely those groups having a partial action, without fixed points, on a finite tree, with a connected globalization. Finally, using Bass-Serre theory, we determine when a partial group action on a graph has a globalization which is a tree.

$Z_2^2$-actions with nnn-dimensional fixed point set

Proceedings of the American Mathematical Society, 2007

We describe the equivariant cobordism classification of smooth actions (M m , Φ) of the group G = Z 2 2 , considered as the group generated by two commuting involutions, on closed smooth m-dimensional manifolds M m , for which the fixed point set of the action is a connected manifold of dimension n and m = 4n − 1 or 4n − 2. For m ≥ 4n, the classification is known. 1991 Mathematics Subject Classification. (2.000 Revision) Primary 57R85; Secondary 57R75. Key words and phrases. Z 2 2-action, fixed data, equivariant cobordism class, characteristic number, projective space bundle, Stiefel-Whitney class. The author was partially supported by CNPq and FAPESP.

Replacement of fixed sets for compact group actions: The 2ρ theorem

2010

If M and N are equivariantly homotopy equivalent G-manifolds, then the fixed sets M^G and N^G are also homotopy equivalent. The replacement problem asks the converse question: If F is homotopy equivalent to the fixed set M^G, is F = N^G for a G-manifold equivariantly homotopy equivalent to M? We prove that for locally linear actions on topological or PL manifolds by compact Lie groups, the replacement is always possible if the normal bundle of the fixed set is twice of a complex bundle over a 1-skeleton of the fixed set. Moreover, we also study some specific examples, where the answer to the replacement problem ranges from always possible to the rigidity.

Acyclic 2-dimensional complexes and Quillen’s conjecture

Publicacions Matemàtiques

Let G be a finite group and Ap(G) be the poset of nontrivial elementary abelian p-subgroups of G. Quillen conjectured that Op(G) is nontrivial if Ap(G) is contractible. We prove that Op(G) = 1 for any group G admitting a G-invariant acyclic p-subgroup complex of dimension 2. In particular, it follows that Quillen's conjecture holds for groups of prank 3. We also apply this result to establish Quillen's conjecture for some particular groups not considered in the seminal work of Aschbacher-Smith.

Replacement of fixed sets for compact group actions: The 2\rho theorem

2009

One fixed point actions on low-dimensional spheres

Inventiones Mathematicae, 1990

When one studies the symmetry groups of spheres, disks and Euclidean spaces, it is often very fruitful to begin by comparing the properties of linear symmetry groups with those of more general examples. In particular, if the fixed point set is finite, it is natural to ask if the number of fixed points coincides with the number for some linear action (namely, 0 or 2 for actions on spheres and 1 for actions on disks and Euclidean spaces). During the nineteen forties P.A. Smith, D. Montgomery and H. Samelson raised questions about the existence of compact Lie group actions on spheres with one fixed point and on disks and Euclidean spaces with no fixed points (see [Eil], Problem 39, and [MSa], Section 7). Recently such actions have attracted additional interest in connection with regularity conjectures for algebraic group actions on affine n-space (see [PR]). Of course, the questions for various spaces are related. In particular, continuous fixed point free actions on Euclidean spaces correspond bijectively to one fixed point actions on spheres via one point compactification. Classical generalizations of Smith theory and the Lefschetz Fixed Point Theorems imply that certain basic families of finite groups can only have the same number of fixed points as linear representations (compare [Gr], [O1]). On the other hand, examples of smooth fixed point free actions on disks and Euclidean spaces exist for groups just outside the ranges of the basic families (e.g., see [FR] and lBrd], Section 1.8). Complete necessary and sufficient conditions for a finite group to act smoothly on disks with no fixed points are given by the work of R. Oliver [O1-O3]. In particular, the smallest finite group admitting such actions is the alternating group A s on five letters. The existence results mentioned above do not give sharp quantitative information on the dimension of the disks and Euclidean spaces that admit fixed point free * The first two authors were partially supported by Louisiana Board of Regents grant RII-8820219, and by NSF Grants DMS-8900878 and DMS-8901583 respectively. The third named author gratefully acknowledges partial support by NSF Grant DMS 86-02543, DMS 89-02622 and the hospitality of the Northwestern University Mathematics Department during part of the preparation of this paper. I. Fixed points of finite group actions on homology 3-spheres Our aim in this part of the paper is to prove the following Theorem I.i. Let M be a compact oriented 3-manifold such that 7tl (M) has no nontrivial representations in SU(2). lf G is a finite group acting smoothly (locally linearly) on M then the number of isolated fixed points either 0 or 2.

Action dimensions of some simple complexes of groups

Journal of Topology, 2019

The action dimension of a discrete group G is the minimum dimension of contractible manifold that admits a proper G-action. We compute the action dimension of the direct limit of a simple complex of groups for several classes of examples including: 1) Artin groups, 2) graph products of groups, and 3) fundamental groups of aspherical complements of arrangements of affine hyperplanes.