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Bruno P. Zimmermann

Publications and source records attributed to Bruno P. Zimmermann.

At least 19 recordsLinked to original sources
arXiv2024

On equivariant embeddings of hyperbolic surfaces into hyperbolic 3-manifolds

We consider the problem of when a closed hyperbolic surface admits a totally geodesic embedding into a closed hyperbolic 3-manifold, and in particular equivariant versions of such embeddings. In a previous paper we considered orientation-preserving actions on orientable surfaces; in the present paper, we consider large orientation-reversing actions on orientable surfaces, and also large actions on nonorientable surfaces.

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arXiv2024

On geodesic embeddings of hyperbolic surfaces into hyperbolic 3-manifolds

We consider the problem of when a closed orientable hyperbolic surface admits a totally geodesic embedding into a closed orientable hyperbolic 3-manifold; given a finite isometric group action on the surface, we consider in particular equivariant versions of such an embedding. We prove that an equivariant embedding exists for all finite irreducible group actions on surfaces; such surfaces are known also as quasiplatonic surfaces; in particular, all quasiplatonic surfaces embed geodesically. In the last section, we discuss some cases of more general finite group actions on surfaces.

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arXiv2024

Large finite group actions on surfaces: Hurwitz groups, maximal reducible and maximal handlebody groups, bounding and non-bounding actions

We consider large finite group-actions on surfaces and discuss and compare various notions for such actions: Hurwitz actions and Hurwitz groups; maximal reducible and completely reducible actions; bounding and geometrically bounding actions; maximal handlebody groups and maximal bounded surface groups; in particular, we discuss small simple groups of various types. A Hurwitz group is a finite group of orientation-preserving diffeomorphisms of maximal possible order $84(g-1)$ of a closed orientable surface of genus $g>1$. A maximal handlebody group instead is a group of orientation-preserving diffeomorphisms of maximal possible order $12(g-1)$ of a 3-dimensional handlebody of genus $g>1$. Among others, we consider the question of when a Hurwitz group acting on a surface of genus $g$ contains a subgroup of maximal possible order $12(g-1)$ extending to a handlebody (or, more generally, a maximal reducible group extending to a product with handles), and show that such Hurwitz groups are closely related to the smallest Hurwitz group ${\rm PSL}_2(7)$ of order 168 acting on Klein's quartic of genus 3. We discuss simple groups of small order which are maximal handlebody groups and, more generaly, maximal reducible groups. We discuss also the problem of which Hurwitz actions bound geometrically, and in particular whether Klein's quartic bounds geometrically: does there exist a compact hyperbolic 3-manifold with totally geodesic boundary isometric to Klein's quartic? Finally, large bounding and non-bounding actions on surfaces of genus 2, 3 and 4 are discussed in section 3.

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arXiv2022

A note on large bounding and non-bounding finite group-actions on surfaces of small genus

The classification of finite group-actions on closed surfaces of small genus is well-known. In the present paper we are interested in the question of which of these group-actions are bounding (extend to a compact 3-manifold with the surface as its unique boundary component, e.g. to a handlebody) or geometrically bounding (extend to a hyperbolic 3-manifold with totally geodesic boundary), concentrating, as a typical case, on large group-actions on surfaces of genus 3.

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arXiv2022

A note on the Nielsen realization problem for connected sums of $S^2 \times S^1$

We consider finite group-actions on 3-manifolds $\cal H_g$ obtained as the connected sum of $g$ copies of $S^2 \times S^1$, with free fundamental group $F_g$ of rank $g$. We prove that, for $g > 1$, a finite group of diffeomorphisms of $\cal H_g$ inducing a trivial action on homology is cyclic. As a consequence, no non-cyclic subgroup of the twist subgroup of the mapping class group of $\cal H_g$ (generated by Dehn twists along embedded 2-spheres) can be realized by diffeomorphisms (in the sense of the Nielsen realization problem). We also discuss when a finite subgroup of the outer automorphism group ${\rm Out}(F_g)$ of the fundamental group of $\cal H_g$ can be realized by a group of diffeomorphisms of $\cal H_g$.

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arXiv2020

Tetrahedral Coxeter groups, large group-actions on 3-manifolds and equivariant Heegaard splittings

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 various manifolds with an action of type 48(g-1) for small values of g, and in particular the unique hyperbolic 3-manifold with such an action of smallest possible genus g = 6 (in strong analogy with the Euclidean case of the 3-torus which has such actions for g = 3).

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arXiv2019

On large orientation-reversing finite group-actions on 3-manifolds and equivariant Heegaard decompositions

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 24(g-1) on a handlebody of genus g. If M is irreducible then it is a spherical, Euclidean or hyperbolic manifold obtained as a quotient of one of the three geometries by a normal subgroup of finite index of a Coxeter group associated to a Coxeter tetrahedron, or of a twisted version of such a Coxeter group.

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arXiv2017

On large groups of symmetries of finite graphs embedded in spheres

Let G be a finite group acting orthogonally on a pair (S^d,Γ) where Γis a finite, connected graph of genus g>1 embedded in the sphere S^d. The 3-dimensional case d=3 has recently been considered in a paper by C. Wang, S. Wang, Y. Zhang and the present author where for each genus g>1 the maximum order of a G-action on a pair (S^3,Γ) is determined and the corresponding graphs Γare classified. In the present paper we consider arbitrary dimensions d and prove that the order of G is bounded above by a polynomial of degree d/2 in g if d is even, and of degree (d+1)/2 if d is odd; moreover the degree d/2 is best possible in even dimensions d. We discuss also the problem, given a finite graph Γand its finite symmetry group, to find the minimal dimension of a sphere into which Γembeds equivariantly as above.

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arXiv2016

On topological actions of finite groups on S^3

We consider orientation-preserving actions of a finite group G on the 3-sphere S^3 (and also on Euclidean space R^3). By the geometrization of finite group actions on 3-manifolds, if such an action is smooth then it is conjugate to an orthogonal action, and in particular G is isomorphic to a subgroup of the orthogonal group SO(4) (or of SO(3) in the case of R^3). On the other hand, there are topological actions with wildly embedded fixed point sets; such actions are not conjugate to smooth actions but one would still expect that the corresponding groups G are isomorphic to subgroups of the orthgonal groups SO(4) (or of SO(3), resp.). In the present paper, we obtain some results in this direction; we prove that the only finite, nonabelian simple group with a topological action on S^3, or on any homology 3-sphere, is the alternating or dodecahedral group A_5 (the only finite, nonabelian simple subgroup of SO(4)), and that every finite group with a topological, orientation-preserving action on Euclidean space R^3 is in fact isomorphic to a subgroup of SO(3).

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arXiv2016

On topological actions of finite, non-standard groups on spheres

The standard actions of finite groups on spheres S^d are linear actions, i.e. by finite subgroups of the orthogonal group O(d+1). We prove that, in each dimension d>5, there is a finite group G which admits a faithful, topological action on a sphere S^d but is not isomorphic to a subgroup of O(d+1). The situation remains open for smooth actions.

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arXiv2015

On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups

It is known that the order of a finite group of diffeomorphisms of a 3-dimensional handlebody of genus g > 1 is bounded by the linear polynomial 12(g-1), and that the order of a finite group of diffeomorphisms of a 4-dimensional handlebody (or equivalently, of its boundary 3-manifold), faithful on the fundamental group, is bounded by a quadratic polynomial in g (but not by a linear one). In the present paper we prove a generalization for handlebodies of arbitrary dimension d, uniformizing handlebodies by Schottky groups and considering finite groups of isometries of such handlebodies. We prove that the order of a finite group of isometries of a handlebody of dimension d acting faithfully on the fundamental group is bounded by a polynomial of degree d/2 in g if d is even, and of degree (d+1)/2 if d is odd, and that the degree d/2 for even d is best possible. This implies then analogous polynomial Jordan-type bounds for arbitrary finite groups of isometries of handlebodies (since a handlebody of dimension d > 3 admits S^1-actions, there does not exist an upper bound for the order of the group itself ).

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arXiv2014

On finite simple groups acting on homology spheres with small fixed point sets

A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets ("pseudofree action") is the alternating group A_5 acting on the 2-sphere. Our first main theorem is the finiteness result that there are only finitely many finite simple groups which admit a smooth action on a homology sphere with at most d-dimensional fixed points sets, for a fixed d. We then go on proving that the finite simple groups acting on a homology sphere with at most 1-dimensional fixed point sets are the alternating group A_5 in dimensions 2, 3 and 5, the linear fractional group PSL_2(7) in dimension 5, and possibly the unitary group PSU_3(3) in dimension 5 (we conjecture that it does not admit any action on a homology 5-sphere but cannot exclude it at present). Finally, we discuss the situation for arbitrary finite groups which admit an action on a homology 3-sphere.

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arXiv2014

On finite groups acting on a connected sum of 3-manifolds S^2 \times S^1

Let H_g denote the closed 3-manifold obtained as the connected sum of g copies of S^2 times S^1, with free fundamental group of rank g. We prove that, for a finite group G acting on H_g which induces a faithful action on the fundamental group, there is an upper bound for the order of G which is quadratic in g, but that there does not exist a linear bound in g. This implies then a Jordan-type bound for arbitrary finite group actions on H_g which is quadratic in g. For the proofs we develop a calculus for finite group-actions on H_g, by codifying such actions by handle-orbifolds and finite graphs of finite groups.

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arXiv2014

On Jordan type bounds for finite groups of diffeomorphisms of 3-manifolds and Euclidean spaces

By a classical result of Jordan, each finite subgroup G of a complex linear group GL_n(C) has an abelian subgroup whose index in G is bounded by a constant depending only on n. We consider the problem if this remains true for finite subgroups G of the diffeomorphism group of a smooth manifold, and show that it is true for all compact 3-manifolds as well as for Euclidean spaces of dimension n < 7. The question remains open at present e.g. for odd-dimensional spheres of dimension greater or equal to five, and for Euclidean spaces of dimension greater or equal to seven.

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arXiv2011

On finite groups acting on spheres and finite subgroups of orthogonal groups

This is a survey on old and new results as well as an introduction to various related basic notions and concepts, based on two talks given at the International Workshop on Geometry and Analysis in Kemerovo (Sobolev Institute of Mathematics, Kemerovo State University) and at the University of Krasnojarsk in June 2011. We discuss finite groups acting on low-dimensional spheres, comparing with the finite subgroups of the corresponding orthogonal groups, and also finite simple groups acting on spheres and homology spheres of arbitrary dimension.

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arXiv2010

On minimal actions of finite simple groups on homology spheres and Euclidean spaces

We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for the linear fractional groups PSL(2,p) as well as for various classes of alternating and symmetric groups. We prove analogous results also for actions on Euclidean spaces.

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