ترغب بنشر مسار تعليمي؟ اضغط هنا

The chain group $C(G)$ of a locally compact group $G$ has one generator $g_{rho}$ for each irreducible unitary $G$-representation $rho$, a relation $g_{rho}=g_{rho}g_{rho}$ whenever $rho$ is weakly contained in $rhootimes rho$, and $g_{rho^*}=g_{rho} ^{-1}$ for the representation $rho^*$ contragredient to $rho$. $G$ satisfies chain-center duality if assigning to each $g_{rho}$ the central character of $rho$ is an isomorphism of $C(G)$ onto the dual $widehat{Z(G)}$ of the center of $G$. We prove that $G$ satisfies chain-center duality if it is (a) a compact-by-abelian extension, (b) connected nilpotent, (c) countable discrete icc or (d) connected semisimple; this generalizes M. M{u}gers result compact groups satisfy chain-center duality.
156 - Thomas Haettel 2021
Starting with a lattice with an action of $mathbb{Z}$ or $mathbb{R}$, we build a Helly graph or an injective metric space. We deduce that the $ell^infty$ orthoscheme complex of any bounded graded lattice is injective. We also prove a Cartan-Hadamard result for locally injective metric spaces. We apply this to show that any Garside group acts on an injective metric space and on a Helly graph. We also deduce that the natural piecewise $ell^infty$ metric on any Euclidean building of type $tilde{A_n}$ extended, $tilde{B_n}$, $tilde{C_n}$ or $tilde{D_n}$ is injective, and its thickening is a Helly graph. Concerning Artin groups of Euclidean types $tilde{A_n}$ and $tilde{C_n}$, we show that the natural piecewise $ell^infty$ metric on the Deligne complex is injective, the thickening is a Helly graph, and it admits a convex bicombing. This gives a metric proof of the $K(pi,1)$ conjecture, as well as several other consequences usually known when the Deligne complex has a CAT(0) metric.
We obtain non-vanishing of group $L^p$-cohomology of Lie groups for $p$ large and when the degree is equal to the rank of the group. This applies both to semisimple and to some suitable solvable groups. In particular, it confirms that Gromovs questio n on vanishing below the rank is formulated optimally. To achieve this, some complementary vanishings are combined with the use of spectral sequences. To deduce the semisimple case from the solvable one, we also need comparison results between various theories for $L^p$-cohomology, allowing the use of quasi-isometry invariance.
We prove an upper bound on the number of pairwise strongly cospectral vertices in a normal Cayley graph, in terms of the multiplicities of its eigenvalues. We use this to determine an explicit bound in Cayley graphs of $mathbb{Z}_2^d$ and $mathbb{Z}_ 4^d$. We also provide some infinite families of Cayley graphs of $mathbb{Z}_2^d$ with a set of four pairwise strongly cospectral vertices and show that such graphs exist in every dimension.
120 - Anthony Genevois 2021
It is known that a cocompact special group $G$ does not contain $mathbb{Z} times mathbb{Z}$ if and only if it is hyperbolic; and it does not contain $mathbb{F}_2 times mathbb{Z}$ if and only if it is toric relatively hyperbolic. Pursuing in this dire ction, we show that $G$ does not contain $mathbb{F}_2 times mathbb{F}_2$ if and only if it is weakly hyperbolic relative to cyclic subgroups, or cyclically hyperbolic for short. This observation motivates the study of cyclically hyperbolic groups, which we initiate in the class of groups acting geometrically on CAT(0) cube complexes. Given such a group $G$, we first prove a structure theorem: $G$ virtually splits as the direct sum of a free abelian group and an acylindrically hyperbolic cubulable group. Next, we prove a strong Tits alternative: every subgroup $H leq G$ either is virtually abelian or it admits a series $H=H_0 rhd H_1 rhd cdots rhd H_k$ where $H_k$ is acylindrically hyperbolic and where $H_i/H_{i+1}$ is finite or free abelian. As a consequence, $G$ is SQ-universal and it cannot contain subgroups such that products of free groups and virtually simple groups.
Existing equivariant neural networks for continuous groups require discretization or group representations. All these approaches require detailed knowledge of the group parametrization and cannot learn entirely new symmetries. We propose to work with the Lie algebra (infinitesimal generators) instead of the Lie group.Our model, the Lie algebra convolutional network (L-conv) can learn potential symmetries and does not require discretization of the group. We show that L-conv can serve as a building block to construct any group equivariant architecture. We discuss how CNNs and Graph Convolutional Networks are related to and can be expressed as L-conv with appropriate groups. We also derive the MSE loss for a single L-conv layer and find a deep relation with Lagrangians used in physics, with some of the physics aiding in defining generalization and symmetries in the loss landscape. Conversely, L-conv could be used to propose more general equivariant ansatze for scientific machine learning.
188 - Bena Tshishiku 2021
We show that finitely-generated, purely pseudo-Anosov subgroups of the genus-2 Goeritz group are convex cocompact in the genus-2 mapping class group.
Coxeter groups are a special class of groups generated by involutions. They play important roles in the various areas of mathematics. This survey particularly focuses on how one use Coxeter groups to construct interesting examples of discrete subgroups of Lie group.
121 - Rod Gow , Gary McGuire 2021
Let $F$ be any field. We give a short and elementary proof that any finite subgroup $G$ of $PGL(2,F)$ occurs as a Galois group over the function field $F(x)$. We also develop a theory of descent to subfields of $F$. This enables us to realize the aut omorphism groups of finite subgroups of $PGL(2,F)$ as Galois groups.
In this paper, we attempt to develop the Quillen Suslin theory for the algebraic fundamental group of a ring. We give a surjective group homomorphism from the algebraic fundamental group of the field of the real numbers to the group of integers. At t he end of the paper, we also propose some problems related to the algebraic fundamental group of some particular type of rings.
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا