arXiv subjects

Bruno Zimmermann

Publications and source records attributed to Bruno Zimmermann.

16 recordsLinked to original sources

arXiv2018

A classification of maximally symmetric surfaces in the 3-dimensional torus

If a finite group of orientation-preserving diffeomorphisms of the 3-dimensional torus leaves invariant an oriented, closed, embedded surface of genus g>1 and preserves the orientation of the surface, then its order is bounded from above by 12(g-1). In the present paper we classify (up to conjugation) all such group actions and surfaces for which the maximal possible order 12(g-1) is achieved, and note that the unknotted surfaces can be realized by equivariant minimal surfaces in a 3-torus.

math.GT↗

arXiv2018

Finite group actions and cyclic branched covers of knots in $\mathbf{S}^3$

We show that a hyperbolic $3$-manifold can be the cyclic branched cover of at most fifteen knots in $\mathbf{S}^3$. This is a consequence of a general result about finite groups of orientation preserving diffeomorphisms acting on $3$-manifolds. A similar, although weaker, result holds for arbitrary irreducible $3$-manifolds: an irreducible $3$-manifold can be the cyclic branched cover of odd prime order of at most six knots in $\mathbf{S}^3$.

math.GT↗

arXiv2017

Graphs in the 3--sphere with maximum symmetry

We consider the orientation-preserving actions of finite groups $G$ on pairs $(S^3, Γ)$, where $Γ$ is a connected graph of genus $g>1$, embedded in $S^3$. For each $g$ we give the maximum order $m_g$ of such $G$ acting on $(S^3, Γ)$ for all such $Γ\subset S^3$. Indeed we will classify all graphs $Γ\subset S^3$ which realize these $m_g$ in different levels: as abstract graphs and as spatial graphs, as well as their group actions. Such maximum orders without the condition "orientation-preserving" are also addressed.

math.GT↗

arXiv2017

Bordered surfaces in the 3-sphere with maximum symmetry

We consider orientation-preserving actions of finite groups $G$ on pairs $(S^3, Σ)$, where $Σ$ denotes a compact connected surface embedded in $S^3$. In a previous paper, we considered the case of closed, necessarily orientable surfaces, determined for each genus $g>1$ the maximum order of such a $G$ for all embeddings of a surface of genus $g$, and classified the corresponding embeddings. In the present paper we obtain analogous results for the case of bordered surfaces $Σ$ (i.e. with non-empty boundary, orientable or not). Now the genus $g$ gets replaced by the algebraic genus $α$ of $Σ$ (the rank of its free fundamental group); for each $α> 1$ we determine the maximum order $m_α$ of an action of $G$, classify the topological types of the corresponding surfaces (topological genus, number of boundary components, orientability) and their embeddings into $S^3$. For example, the maximal possibility $12(α- 1)$ is obtained for the finitely many values $α= 2, 3, 4, 5, 9, 11, 25, 97, 121$ and $241$.

math.GT↗

arXiv2017

Embedding compact surfaces into the 3-dimensional Euclidean space with maximum symmetry

The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, Σ)$ among all possible embedded closed/bordered surfaces with given geometric/algebraic genus $>1$ in $\mathbb{R}^3$. We also identify the topological types of the bordered surfaces realizing the maximum order, and find simple representative embeddings for such surfaces.

math.GT↗

arXiv2016

Embedding surfaces into $S^3$ with maximum symmetry

We restrict our discussion to the orientable category. For $g > 1$, let $OE_g$ be the maximum order of a finite group $G$ acting on the closed surface $Σ_g$ of genus $g$ which extends over $(S^3, Σ_g)$, where the maximum is taken over all possible embeddings $Σ_g\hookrightarrow S^3$. We will determine $OE_g$ for each $g$, indeed the action realizing $OE_g$. In particular, with 23 exceptions, $OE_g$ is $4(g+1)$ if $g\ne k^2$ or $4(\sqrt{g}+1)^2$ if $g=k^2$, and moreover $OE_g$ can be realized by unknotted embeddings for all $g$ except for $g=21$ and $481$.

math.GT↗

arXiv2012

Extending finite group actions on surfaces over $S^3$

Let $OE_g$ (resp. $CE_g$ and $AE_g$) and resp. $OE^o_g$ be the maximum order of finite (resp. cyclic and abelian) groups $G$ acting on the closed orientable surfaces $Σ_g$ which extend over $(S^3, Σ_g)$ among all embeddings $Σ_g\to S^3$ and resp. unknotted embeddings $Σ_g\to S^3$. It is known that $OE^o_g\le 12(g-1)$, and we show that $12(g-1)$ is reached for an unknotted embedding $Σ_g \to S^3$ if and only if $g = 2$, 3, 4, 5, 6, 9, 11, 17, 25, 97, 121, 241, 601. Moreover $AE_g$ is $2g+2$; and $CE_g$ is $2g+2$ for even $g$, and $2g-2$ for odd $g$. Efforts are made to see intuitively how these maximal symmetries are embedded into the symmetries of the 3-sphere.

math.GT↗

arXiv2011

On finite simple groups acting on homology spheres

It is a consequence of the classical Jordan bound for finite subgroups of linear groups that in each dimension n there are only finitely many finite simple groups which admit a faithful, linear action on the n-sphere. In the present paper we prove an analogue for smooth actions on arbitrary homology n-spheres: in each dimension n there are only finitely many finite simple groups which admit a faithful, smooth action on some homology sphere of dimension n, and in particular on the n-sphere. We discuss also the finite simple groups which admit an action on a homology sphere of dimension 3, 4 or 5.

math.GT↗

arXiv2011

High-resolution imaging of ultracold fermions in microscopically tailored optical potentials

We report on the local probing and preparation of an ultracold Fermi gas on the length scale of one micrometer, i.e. of the order of the Fermi wavelength. The essential tool of our experimental setup is a pair of identical, high-resolution microscope objectives. One of the microscope objectives allows local imaging of the trapped Fermi gas of 6Li atoms with a maximum resolution of 660 nm, while the other enables the generation of arbitrary optical dipole potentials on the same length scale. Employing a 2D acousto-optical deflector, we demonstrate the formation of several trapping geometries including a tightly focussed single optical dipole trap, a 4x4-site two-dimensional optical lattice and a 8-site ring lattice configuration. Furthermore, we show the ability to load and detect a small number of atoms in these trapping potentials. A site separation of down to one micrometer in combination with the low mass of 6Li results in tunneling rates which are sufficiently large for the implementation of Hubbard-models with the designed geometries.

cond-mat.quant-gas↗

arXiv2011

On minimal finite quotients of outer automorphism groups of free groups

We prove that, for n=3 and 4, the minimal nonabelian finite factor group of the outer automorphism group Out F_n of a free group of rank n is the linear group PSL_n(Z_2) (conjecturally, this may remain true for arbitrary rank n > 2). We also discuss some computational results on low index subgroups of Aut F_n and Out F_n, for n = 3 and 4, using presentations of these groups.

math.GR↗

arXiv2010

On finite groups acting on acyclic low-dimensional manifolds

We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively. The analogue remains open in dimension five (where it is not true for arbitrary continuous actions, however). We prove that the only finite nonabelian simple groups admitting a smooth action on an acyclic 5-manifold are the alternating groups A_5 and A_6, and deduce from this a short list of finite groups, closely related to the finite subgroups of SO(5), which are the candidates for orientation-preserving actions on acyclic 5-manifolds.

math.GT↗

arXiv2010

On finite groups acting on homology 4-spheres and finite subgroups of SO(5)

We show that a finite group which admits a faithful, smooth, orientation-preserving action on a homology 4-sphere, and in particular on the 4-sphere, is isomorphic to a subgroup of the orthogonal group SO(5), by explicitly determining the various groups which can occur (up to an indetermination of index two in the case of solvable groups). As a consequence we obtain also a characterization of the finite groups which are isomorphic to subgroups of the orthogonal groups SO(5) and O(5).

math.GT↗

arXiv2009

Analog control of open quantum systems under arbitrary decoherence

We derive and investigate a general non-Markovian equation for the time-dependence of a Hamiltonian that maximizes the fidelity of a desired quantum gate on any finite-dimensional quantum system in the presence of arbitrary bath and noise sources. The method is illustrated for a single-qubit gate implemented on a three-level system.

quant-ph↗

arXiv2009

A characterisation of S^3 among homology spheres

We prove that an integral homology 3-sphere is S^3 if and only if it admits four periodic diffeomorphisms of odd prime orders whose space of orbits is S^3. As an application we show that an irreducible integral homology sphere which is not S^3 is the cyclic branched cover of odd prime order of at most four knots in S^3. A result on the structure of finite groups of odd order acting on integral homology spheres is also obtained.

math.GT↗

arXiv2008

On finite simple and nonsolvable groups acting on closed 4-manifolds

We show that the only finite nonabelian simple groups which admit a locally linear, homologically trivial action on a closed simply connected 4-manifold $M$ (or on a 4-manifold with trivial first homology) are the alternating groups $A_5$, $A_6$ and the linear fractional group PSL(2,7) (we note that for homologically nontrivial actions all finite groups occur). The situation depends strongly on the second Betti number $b_2(M)$ of $M$ and has been known before if $b_2(M)$ is different from two, so the main new result of the paper concerns the case $b_2(M)=2$. We prove that the only simple group that occurs in this case is $A_5$, and then give a short list of finite nonsolvable groups which contains all candidates for actions of such groups.

math.GT↗

arXiv2005

On finite simple and nonsolvable groups acting on homology 4-spheres

The only finite nonabelian simple group acting on a homology 3-sphere - necessarily non-freely - is the dodecahedral group $\Bbb A_5 \cong {\rm PSL}(2,5)$ (in analogy, the only finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group $\Bbb A_5^* \cong {\rm SL}(2,5)$). In the present paper we show that the only finite simple groups acting on a homology 4-sphere, and in particular on the 4-sphere, are the alternating or linear fractional groups groups $\Bbb A_5 \cong {\rm PSL}(2,5)$ and $\Bbb A_6 \cong {\rm PSL}(2,9)$. From this we deduce a short list of groups which contains all finite nonsolvable groups admitting an action on a homology 4-spheres.

math.GT↗