We examine free orientation-reversing group actions on orientable handlebodies, and free actions on nonorientable handlebodies. A classification theorem is obtained, giving the equivalence classes and weak equivalence classes of free actions in terms of algebraic invariants that involve Nielsen equivalence. This is applied to describe the sets of free actions in various cases, including a complete classification for many (and conjecturally all) cases above the minimum genus. For abelian groups, the free actions are classified for all genera.
Fix a free, orientation-preserving action of a finite group G on a 3-dimensional handlebody V. Whenever G acts freely preserving orientation on a connected 3-manifold X, there is a G-equivariant imbedding of V into X. There are choices of X closed and Seifert-fibered for which the image of V is a handlebody of a Heegaard splitting of X. Provided that the genus of V is at least 2, there are similar choices with X closed and hyperbolic.
We study the orientation preserving involutions of the orientable 3-dimensional handlebody $H_g$, for any genus $g$. A complete classification of such involutions is given in terms of their fixed points.
An extended Kleinian group whose orientation-preserving half is a Schottky group is called an extended Schottky group. These groups correspond to the real points in the Schottky space. Their geometric structures is well known and it permits to provide information on the locus of fixed points of symmetries of handlebodies. A group generated by two different extended Schottky groups, both with the same orientation-preserving half, is called a dihedral extended Schottky group. We provide a structural description of these type of groups and, as a consequence, we obtain sharp upper bounds for the s…
The aim of this note is to stablish the topological classification of finite period orientation reversing autohomeomorphims of a closed oriented surface when the period is 2q, with q even. The classification of periodic orientation reversing autohomeomorphims of a closed oriented surface has been made by Kazuo Yokoyama in Complete classification of periodic maps on compact surfaces, Tokyo J. Math. 15 (1992), no. 2, 247--279, and by the author in Classification of the orientation reversing homeomorphisms of finite order of surfaces, Topology and its applications 62 (1995) 145--162 (reference [1…
Given a symmetry $τ$ of a closed Riemann surface $S$, there exists an extended Kleinian group $K$, whose orientation-preserving half is a Schottky group $Γ$ uniformizing $S$, such that $K/Γ$ induces $\langle τ\rangle$; the group $K$ is called an extended Schottky group. A geometrical structural description, in terms of the Klein-Maskit combination theorems, of both Schottky and extended Schottky groups is well known. A dihedral extended Schottky group is a group generated by the elements of two different extended Schottky groups, both with the same orientation-preserving half. Such configurati…
We present a new description of the genus 3 Arnoux--Yoccoz translation surface in terms of its Delaunay polygons and show that, up to affine equivalence, it belongs to two families of surfaces whose isometry groups include the dihedral group of the square.
We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus g > 1 is 24(g-1), and in the present paper we characterize the 3-manifolds M and groups G for which the maximal possible order |G| = 24(g-1) is obtained, for some G-invariant Heegaard splitting of genus g > 1. If M is reducible then it is obtained by doubling an action of maximal possible order 2…
In this paper we consider all orientation-preserving $\mathbb{Z}_{4}$-actions on $3$-dimensional handlebodies $V_g$ of genus $g>0$. We study the graph of groups $(Γ($v$),\mathbf{G(v)})$, which determines a handlebody orbifold $V(Γ($v$),{\mathbf{G(v)}})\simeq{V_g}/\mathbb{Z}_{4}$. This algebraic characterization is used to enumerate the total number of $\mathbb{Z}_{4}$ group actions on such handlebodies, up to equivalence.
In this paper, we give a classification of orientation reversing periodic maps on closed surfaces which generalizes the theory of Nielsen for the orientation preserving periodic maps. On one hand, we give a group of data for each orientation reversing periodic map such that two periodic maps with the same data must be conjugate to each other. On the other hand, we give the criterion to judge when two different groups of data correspond to the same conjugacy class. As an application of the results of this paper, we shall show that a given orientation reversing periodic map on $Σ_g$ with period …
The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable 3-dimensional handlebody of genus g can be enumerated in terms of sets of generators of G. They correspond to the equivalence classes of generating n-vectors of elements of G, where n=1+(g-1)/|G|, under Nielsen equivalence (or weak Nielsen equivalence). For abelian and dihedral G, this allows a complete determination of the equivalence and weak equivalence classes of actions for all genera. Additional information is obtained for solvable groups and for the groups PSL(2,3^p)…
We give a geometric characterization of compact Riemann surfaces admitting orientation reversing involutions with fixed points. Such surfaces are generally called real surfaces and can be represented by real algebraic curves with non-empty real part. We show that there is a family of disjoint simple closed geodesics that intersect all geodesics of a partition at least twice in uniquely right angles if and only if the involution exists. This implies that a surface is real if and only if there is a pants decomposition of the surface with all Fenchel-Nielsen twist parameters equal to 0 or 1/2.
Let $\textup{H}_g$ be a genus $g$ handlebody and $\textup{MCG}_{2n}(\textup{T}_g)$ be the group of the isotopy classes of orientation preserving homeomorphisms of $\textup{T}_g=\partial\textup{H}_g$, fixing a given set of $2n$ points. In this paper we find a finite set of generators for $\mathcal{E}_{2n}^g$, the subgroup of $\textup{MCG}_{2n}(\textup{T}_g)$ consisting of the isotopy classes of homeomorphisms of $\textup{T}_g$ admitting an extension to the handlebody and keeping fixed the union of $n$ disjoint properly embedded trivial arcs. This result generalizes a previous one obtained by th…
Let $M$ be a connected compact orientable surface, $f:M\to \mathbb{R}$ be a Morse function, and $h:M\to M$ be a diffeomorphism which preserves $f$ in the sense that $f\circ h = f$. We will show that if $h$ leaves invariant each regular component of each level set of $f$ and reverses its orientation, then $h^2$ is isotopic to the identity map of $M$ via $f$-preserving isotopy. This statement can be regarded as a foliated and a homotopy analogue of a well known observation that every reversing orientation orthogonal isomorphism of a plane has order $2$, i.e. is a mirror symmetry with respect to …
We consider finite group-actions on closed, orientable and nonorientable 3-manifolds M which preserve the two handlebodies of a Heegaard splitting of M of some genus g > 1 (maybe interchanging the two handlebodies). The maximal possible order of a finite group-action on a handlebody of genus g>1 is 12(g-1) in the orientation-preserving case and 24(g-1) in general, and the maximal order of a finite group preserving the Heegaard surface of a Heegaard splitting of genus g is 48(g-1). This defines a hierarchy for finite group-actions on 3-manifolds which we discuss in the present paper; we present…
This paper discusses topological and locally linear actions of finite groups on $S^4$. Local linearity of the orientation preserving actions on $S^4$ forces the group to be a subgroup of $SO(5)$. On the other hand, orientation reversing topological actions of "exotic" groups $G$ (i.e. $G\not\subset O(5)$) on $S^4$ are constructed, and local linearity and stable smoothability of the actions are studied.
R. S. Kulkarni showed that a finite group acting pseudofreely, but not freely, preserving orientation, on an even-dimensional sphere (or suitable sphere-like space) is either a periodic group acting semifreely with two fixed points, a dihedral group acting with three singular orbits, or one of the polyhedral groups, occurring only in dimension 2. It is shown here that the dihedral group does not act pseudofreely and locally linearly on an actual n-sphere when n is congruent to 0 mod 4. The possibility of such an action when n is congruent to 2 mod 4 and n>2 remains open. Orientation-reversing …
In two previous papers the author presented a general construction of finite, fiber- and orientation-preserving group actions on orientable Seifert manifolds. In this paper we restrict our attention to elliptic 3-manifolds. A proof is given that orientation-reversing and fiber-preserving diffeomorphisms of Seifert manifolds do not exist for nonzero Euler class, in particular elliptic 3-manifolds. Each type of elliptic 3-manifold is then considered and the possible group actions that fit the given construction. This is shown to be all but a few cases that have been considered elsewhere. Finally…
Let {a,b} and {c,d} be two pairs of bounding simple closed curves on an oriented surface which intersect nontrivialy. We prove that if these pairs are invariant under the action of an orientation reversing involution, then the corresponding bounding pair maps generate a free group. This supports the conjecture stated by C. Leininger and D. Margalit that any pair of elements of the Torelli group either commute or generate a free group.
The standard P. A. Smith theory of p-group actions on spheres, disks, and euclidean spaces is extended to the case of p-group actions on tori (i.e., products of circles) and coupled with topological surgery theory to give a complete topological classification, valid in all dimensions, of the locally linear, orientation reversing, involutions on tori with fixed point set of codimension one.
We present a complete classification of elements in the mapping class group of the torus which have a representative that can be written as a product of two orientation reversing involutions. Our interest in such decompositions is motivated by features of the monodromy maps of real fibrations. We employ the property that the mapping class group of the torus is identifiable with $SL(2,\Z)$ as well as that the quotient group $PSL(2,\Z)$ is the symmetry group of the {\em Farey tessellation} of the Poincaré disk.
We present a new theory which describes the collection of all tunnels of tunnel number 1 knots in the 3-sphere (up to orientation-preserving equivalence in the sense of Heegaard splittings) using the disk complex of the genus-2 handlebody and associated structures. It shows that each knot tunnel is obtained from the tunnel of the trivial knot by a uniquely determined sequence of simple cabling constructions. A cabling construction is determined by a single rational parameter, so there is a corresponding numerical parameterization of all tunnels by sequences of such parameters and some addition…
In [Tohoku Math. J. 62 (2010), 45--53] the second author showed that, except for a few cases, the order $N$ of a cyclic group of self-homeomorphisms of a closed orientable topological surface $S_g$ of genus $g \geq 2$ determines the group up to a topological conjugation, provided that $N\geq 3g$. The first author et al. undertook in [Collect. Math. 67 (2016), 415--429] a more general problem of topological classification of such group actions for $N>2(g-1)$. In [Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A. Mat. (RACSAM) 110 (2016), 303--320] we considered the analogous problem for closed no…
