1 - Oberwolfach Preprints (OWP)
http://publications.mfo.de/handle/mfo/19
The Oberwolfach Preprints (OWP) mainly contain research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs program and the Oberwolfach Leibniz Fellows, but this can also include an Oberwolfach Lecture, for example.Tue, 25 Jun 2019 05:21:46 GMT2019-06-25T05:21:46ZOn Residuals of Finite Groups
http://publications.mfo.de/handle/mfo/1423
On Residuals of Finite Groups
Aivazidis, Stefanos; Müller, Thomas
A theorem of Dolfi, Herzog, Kaplan, and Lev [DHKL07, Thm. C] asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in [DHKL07], we establish the following generalisation: if ${\mathfrak{X}}$ is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and $\overline{\mathfrak{X}}$ is the extension-closure of $\mathfrak{X}$, then there exists an (optimal) constant $\gamma$ depending only on $\mathfrak{X}$ such that, for all non-trivial finite groups G with trivial $\mathfrak{X}$-radical, ${\vert}G{\vert}^{\overline{\mathfrak{X}}} > {\vert}G{\vert}^\gamma$, where $G^{\overline{\mathfrak{X}}}$ is the ${\overline{\mathfrak{X}}}$-residual of $G$. When ${\mathfrak{X}}={\mathfrak{N}}$, the class of finite nilpotent groups, it follows that $\overline{\mathfrak{X}} = \mathfrak{S}$, the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson's classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations X of full characteristic such that $\mathfrak{S}
\subset \overline{\mathfrak{X}} \subset \mathfrak{E}$, thus providing applications of our main result beyond the reach of [DHKL07, Thm. C]
Tue, 28 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14232019-05-28T00:00:00ZAivazidis, StefanosMüller, ThomasA theorem of Dolfi, Herzog, Kaplan, and Lev [DHKL07, Thm. C] asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in [DHKL07], we establish the following generalisation: if ${\mathfrak{X}}$ is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and $\overline{\mathfrak{X}}$ is the extension-closure of $\mathfrak{X}$, then there exists an (optimal) constant $\gamma$ depending only on $\mathfrak{X}$ such that, for all non-trivial finite groups G with trivial $\mathfrak{X}$-radical, ${\vert}G{\vert}^{\overline{\mathfrak{X}}} > {\vert}G{\vert}^\gamma$, where $G^{\overline{\mathfrak{X}}}$ is the ${\overline{\mathfrak{X}}}$-residual of $G$. When ${\mathfrak{X}}={\mathfrak{N}}$, the class of finite nilpotent groups, it follows that $\overline{\mathfrak{X}} = \mathfrak{S}$, the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson's classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations X of full characteristic such that $\mathfrak{S}
\subset \overline{\mathfrak{X}} \subset \mathfrak{E}$, thus providing applications of our main result beyond the reach of [DHKL07, Thm. C]Congruences Associated with Families of Nilpotent Subgroups and a Theorem of Hirsch
http://publications.mfo.de/handle/mfo/1422
Congruences Associated with Families of Nilpotent Subgroups and a Theorem of Hirsch
Aivazidis, Stefanos; Müller, Thomas
Our main result associates a family of congruences with each suitable system of nilpotent subgroups of a finite group. Using this result, we complete and correct the proof of a theorem of Hirsch concerning the class number of a finite group of odd order.
Mon, 27 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14222019-05-27T00:00:00ZAivazidis, StefanosMüller, ThomasOur main result associates a family of congruences with each suitable system of nilpotent subgroups of a finite group. Using this result, we complete and correct the proof of a theorem of Hirsch concerning the class number of a finite group of odd order.Experimenting with Symplectic Hypergeometric Monodromy Groups
http://publications.mfo.de/handle/mfo/1420
Experimenting with Symplectic Hypergeometric Monodromy Groups
Detinko, Alla; Flannery, Dane; Hulpke, Alexander
We present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying arithmeticity. The results are obtained by extending our previous algorithms for Zariski dense groups, based on the strong approximation and congruence subgroup properties.
Wed, 22 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14202019-05-22T00:00:00ZDetinko, AllaFlannery, DaneHulpke, AlexanderWe present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying arithmeticity. The results are obtained by extending our previous algorithms for Zariski dense groups, based on the strong approximation and congruence subgroup properties.Chirality of Real Non-Singular Cubic Fourfolds and Their Pure Deformation Classification
http://publications.mfo.de/handle/mfo/1419
Chirality of Real Non-Singular Cubic Fourfolds and Their Pure Deformation Classification
Finashin, Sergey; Kharlamov, Viatcheslav
In our previous works we have classified real non-singular cubic hypersurfaces in the 5-dimensional projective space up to equivalence that includes both real projective transformations and continuous variations of co-efficients preserving the hypersurface non-singular. Here, we perform a finer classification giving a full answer to the chirality problem: which of real non-singular cubic hypersurfaces can not be continuously deformed to their mirror reflection.
Wed, 15 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14192019-05-15T00:00:00ZFinashin, SergeyKharlamov, ViatcheslavIn our previous works we have classified real non-singular cubic hypersurfaces in the 5-dimensional projective space up to equivalence that includes both real projective transformations and continuous variations of co-efficients preserving the hypersurface non-singular. Here, we perform a finer classification giving a full answer to the chirality problem: which of real non-singular cubic hypersurfaces can not be continuously deformed to their mirror reflection.The Becker-Gottlieb Transfer: a Geometric Description
http://publications.mfo.de/handle/mfo/1418
The Becker-Gottlieb Transfer: a Geometric Description
Wang, Yi-Sheng
In this note, we examine geometric aspects of the Becker-Gottlieb transfer in terms of the Umkehr and index maps, and rework some classic index theorems, using the cohomological formulae of the Becker-Gottlieb transfer. The results are natural from the homotopy-theoretic point of view; they reveal subtle geometric information in the Umkehr map, and demonstrate the beauty of the Atiyah-Singer index theorem for families and its generalizations.
Tue, 14 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14182019-05-14T00:00:00ZWang, Yi-ShengIn this note, we examine geometric aspects of the Becker-Gottlieb transfer in terms of the Umkehr and index maps, and rework some classic index theorems, using the cohomological formulae of the Becker-Gottlieb transfer. The results are natural from the homotopy-theoretic point of view; they reveal subtle geometric information in the Umkehr map, and demonstrate the beauty of the Atiyah-Singer index theorem for families and its generalizations.The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth
http://publications.mfo.de/handle/mfo/1417
The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth
Biswas, Kingshook; Knieper, Gerhard; Peyerimhoff, Norbert
Let $X$ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except for the flat spaces. Denote by $h > 0$ the mean curvature of horospheres in $X$, and set $\rho = h/2$. Fixing a basepoint $o \in X$, for $\xi \in \partial X$, denote by $B_{\xi}$ the Busemann function at $\xi$ such that $B_{\xi}(o) = 0$. then for $\lambda \in \mathbb{C}$ the function $e^{(i\lambda - \rho)B_{\xi}}$ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue $-(\lambda^2 + \rho^2)$. For a function $f$ on $X$, we define the Fourier transform of $f$ by $$\tilde{f}(\lambda, \xi) := \int_X f(x) e^{(-i\lambda - \rho)B_{\xi}(x)} dvol(x)$$ for all $\lambda \in \mathbb{C}, \xi \in \partial X$ for which the integral converges. We prove a Fourier inversion formula $$f(x) = C_0 \int_{0}^{\infty} \int_{\partial X}
\tilde{f}(\lambda, \xi) e^{(i\lambda - \rho)B_{\xi}(x)} d\lambda_o(\xi) |c(\lambda)|^{-2} d\lambda$$ for $f \in C^{\infty}_c(X)$, where $c$ is a certain function on $\mathbb{R} - \{0\}$, $\lambda_o$ is the visibility measure on $\partial X$ with respect to the basepoint $o \in X$ and $C_0 > 0$ is a constant. We also prove a Plancherel theorem, and a version of the Kunze-Stein phenomenon.
Wed, 08 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14172019-05-08T00:00:00ZBiswas, KingshookKnieper, GerhardPeyerimhoff, NorbertLet $X$ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except for the flat spaces. Denote by $h > 0$ the mean curvature of horospheres in $X$, and set $\rho = h/2$. Fixing a basepoint $o \in X$, for $\xi \in \partial X$, denote by $B_{\xi}$ the Busemann function at $\xi$ such that $B_{\xi}(o) = 0$. then for $\lambda \in \mathbb{C}$ the function $e^{(i\lambda - \rho)B_{\xi}}$ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue $-(\lambda^2 + \rho^2)$. For a function $f$ on $X$, we define the Fourier transform of $f$ by $$\tilde{f}(\lambda, \xi) := \int_X f(x) e^{(-i\lambda - \rho)B_{\xi}(x)} dvol(x)$$ for all $\lambda \in \mathbb{C}, \xi \in \partial X$ for which the integral converges. We prove a Fourier inversion formula $$f(x) = C_0 \int_{0}^{\infty} \int_{\partial X}
\tilde{f}(\lambda, \xi) e^{(i\lambda - \rho)B_{\xi}(x)} d\lambda_o(\xi) |c(\lambda)|^{-2} d\lambda$$ for $f \in C^{\infty}_c(X)$, where $c$ is a certain function on $\mathbb{R} - \{0\}$, $\lambda_o$ is the visibility measure on $\partial X$ with respect to the basepoint $o \in X$ and $C_0 > 0$ is a constant. We also prove a Plancherel theorem, and a version of the Kunze-Stein phenomenon.Minimal Codimension One Foliation of a Symmetric Space by Damek-Ricci Spaces
http://publications.mfo.de/handle/mfo/1416
Minimal Codimension One Foliation of a Symmetric Space by Damek-Ricci Spaces
Knieper, Gerhard; Parker, John R.; Peyerimhoff, Norbert
In this article we consider solvable hypersurfaces of the form $N \exp(\mathbb{R} H)$ with induced metrics in the symmetric space $M = SL(3,\mathbb{C})/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition $SL(3,\mathbb{C}) = NAK$. Since $M$ is rank $2$, $A$ is $2$-dimensional and we can parametrize these hypersurfaces via an angle $\alpha \in [0,\pi/2]$ determining the direction of $H$. We show that one of the hypersurfaces (corresponding to $\alpha = 0$) is minimally embedded and isometric to the non-symmetric $7$-dimensional Damek-Ricci space. We also provide an explicit formula for the
Ricci curvature of these hypersurfaces and show that all hypersurfaces for $\alpha \in (0,\frac{\pi}{2}]$ admit planes of both negative and positive sectional curvature. Moreover, the symmetric space $M$ admits a minimal foliation with all leaves isometric to the non-symmetric $7$-dimensional Damek-Ricci space.
Tue, 07 May 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14162019-05-07T00:00:00ZKnieper, GerhardParker, John R.Peyerimhoff, NorbertIn this article we consider solvable hypersurfaces of the form $N \exp(\mathbb{R} H)$ with induced metrics in the symmetric space $M = SL(3,\mathbb{C})/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition $SL(3,\mathbb{C}) = NAK$. Since $M$ is rank $2$, $A$ is $2$-dimensional and we can parametrize these hypersurfaces via an angle $\alpha \in [0,\pi/2]$ determining the direction of $H$. We show that one of the hypersurfaces (corresponding to $\alpha = 0$) is minimally embedded and isometric to the non-symmetric $7$-dimensional Damek-Ricci space. We also provide an explicit formula for the
Ricci curvature of these hypersurfaces and show that all hypersurfaces for $\alpha \in (0,\frac{\pi}{2}]$ admit planes of both negative and positive sectional curvature. Moreover, the symmetric space $M$ admits a minimal foliation with all leaves isometric to the non-symmetric $7$-dimensional Damek-Ricci space.On the Lie Algebra Structure of $HH^1(A)$ of a Finite-Dimensional Algebra A
http://publications.mfo.de/handle/mfo/1412
On the Lie Algebra Structure of $HH^1(A)$ of a Finite-Dimensional Algebra A
Linckelmann, Markus; Rubio y Degrassi, Lleonard
Let $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main result shows that if the Ext-quiver of $A$ has no loops and at most two parallel arrows in any direction, and if $HH^1(A)$ is a simple Lie algebra, then char(k) is not equal to $2$ and $HH^1(A)\cong$ $sl_2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.
Wed, 17 Apr 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14122019-04-17T00:00:00ZLinckelmann, MarkusRubio y Degrassi, LleonardLet $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main result shows that if the Ext-quiver of $A$ has no loops and at most two parallel arrows in any direction, and if $HH^1(A)$ is a simple Lie algebra, then char(k) is not equal to $2$ and $HH^1(A)\cong$ $sl_2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.The First Hochschild Cohomology as a Lie Algebra
http://publications.mfo.de/handle/mfo/1411
The First Hochschild Cohomology as a Lie Algebra
Rubio y Degrassi, Lleonard; Schroll, Sibylle; Solotar, Andrea
In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvabilty of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including quantum complete intersections.
Tue, 16 Apr 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14112019-04-16T00:00:00ZRubio y Degrassi, LleonardSchroll, SibylleSolotar, AndreaIn this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvabilty of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including quantum complete intersections.On a Group Functor Describing Invariants of Algebraic Surfaces
http://publications.mfo.de/handle/mfo/1409
On a Group Functor Describing Invariants of Algebraic Surfaces
Dietrich, Heiko; Moravec, Primož
Liedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $\tilde K$ to a group functor $\tau$ arising in the construction of the non-abelian exterior square of a group. In contrast to $\tilde K$, there exist efficient algorithms for constructing $\tau$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $\tau(G)$, and when $\tau(G)$ and $\tilde K(G,3)$ are isomorphic.
Fri, 01 Mar 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14092019-03-01T00:00:00ZDietrich, HeikoMoravec, PrimožLiedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $\tilde K$ to a group functor $\tau$ arising in the construction of the non-abelian exterior square of a group. In contrast to $\tilde K$, there exist efficient algorithms for constructing $\tau$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $\tau(G)$, and when $\tau(G)$ and $\tilde K(G,3)$ are isomorphic.