Abstract homomorphisms of algebraic groups and applications

The Number of Group Homomorphisms from $D_m$ into $D_n$

Counting homomorphisms between cyclic groups is a common exercise in a first course in abstract algebra. A similar problem, accessible at the same level, is to count the number of group homomorphisms from a dihedral group of order $2m$ into a dihedral group of order $2n$. While the solution requires only elementary group theory, the result does not appear in the literature or in the usual texts. As the solution may be of interest, particularly to those teaching undergraduate abstract algebra, it is provided in this note.

Delooping the sign homomorphism in univalent mathematics

In univalent mathematics there are at least two equivalent ways to present the category of groups. Groups presented in their usual algebraic form are called abstract groups, and groups presented as pointed connected $1$-types are called concrete groups. Since these two descriptions of the category of groups are equivalent, we find that every algebraic group corresponds uniquely to a concrete group -- its delooping -- and that each abstract group homomorphisms corresponds uniquely to a pointed map between concrete groups. The $n$-th abstract symmetric group $S_n$ of all bijections $[n]\simeq [n…

On a question of Külshammer for homomorphisms of algebraic groups

Let $G$ be a linear algebraic group over an algebraically closed field of characteristic $p\geq 0$. We show that if $H_1$ and $H_2$ are connected subgroups of $G$ such that $H_1$ and $H_2$ have a common maximal unipotent subgroup and $H_1/R_u(H_1)$ and $H_2/R_u(H_2)$ are semisimple, then $H_1$ and $H_2$ are $G$-conjugate. Moreover, we show that if $H$ is a semisimple linear algebraic group with maximal unipotent subgroup $U$ then for any algebraic group homomorphism $σ\colon U\rightarrow G$, there are only finitely many $G$-conjugacy classes of algebraic group homomorphisms $ρ\colon H\rightarr…

从某些拓扑群到柱形双曲群的抽象同态

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

该研究刻画了从完全可度量化或局部可数紧豪斯多夫拓扑群到柱形双曲群的同态性质，得到映射类群、单端双曲群外自同构群相关的精确结果，推导了相对双曲群等的自动连续性，还证明夏威夷耳环群是柱形双曲群但无通用柱形作用。

关于单代数群中格的拟同态刚性

On quasi-homomorphism rigidity for lattices in simple algebraic groups

本文改进了Ozawa的结果，证明了局部域上秩$\geq 2$的单代数群及第二可数局部紧群中的格具有property $(TTT)$，并探讨了具有无限中心的Lie群对该性质的缺失。

到实代数群的准同态

Quasihomomorphisms to real algebraic groups

本文研究了实线性代数群中的准同态，证明了与离散群中类似但针对实代数群的刚性定理。

Splitting homomorphisms and the Geometrization Conjecture

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture, and the other is equivalent to the union of the Geometrization Conjecture and Thurston's Virtual Bundle Conjecture.

On abstract representations of the groups of rational points of algebraic groups and their deformations

In this paper, we continue our study of abstract representations of elementary subgroups of Chevalley groups of rank $\geq 2.$ First, we extend our earlier methods to analyze representations of elementary groups over arbitrary associative rings, and as a consequence, prove the conjecture of Borel and Tits on abstract homomorphisms of the groups of rational points of algebraic groups for groups of the form ${\bf SL}_{n,D}$, where $D$ is a finite-dimensional central division algebra over a field of characteristic zero. Second, we apply our results to study deformations of representations of elem…

通过对称性和同态多项式进行代数复杂性下的下界

Lower Bounds in Algebraic Complexity via Symmetry and Homomorphism Polynomials

本文通过引入对称代数复杂性类，证明了对称代数电路的下界，并扩展了同态多项式在复杂性类中的完整性和下界分析。

Abstract commensurators of solvable Baumslag - Solitar groups

We prove that for any natural n>1, the abstract commensurator group of the Baumslag - Solitar group BS(1,n) is isomorphic to the group of 2 by 2 upper triangular matrices A over rational numbers with A_{11}=1. We also prove that for any finitely generated group G with the unique root property the natural homomorphisms Aut(G)--> Comm(G)--> QI(G) are embeddings.

Homomorphic images of algebraic groups

We study topological group theoretic properties of algebraic groups over local fields. In particular, we find conditions under which such groups have closed images under arbitrary continuous homomorphisms into arbitrary topological groups.

Abstract group actions of locally compact groups on CAT(0) spaces

We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion free CAT(0) group is continuous.

Injective homomorphisms of mapping class groups of non-orientable surfaces

Let $N$ be a compact, connected, non-orientable surface of genus $ρ$ with $n$ boundary components, with $ρ\ge 5$ and $n \ge 0$, and let $\mathcal{M} (N)$ be the mapping class group of $N$. We show that, if $\mathcal{G}$ is a finite index subgroup of $\mathcal{M} (N)$ and $φ: \mathcal{G} \to \mathcal{M} (N)$ is an injective homomorphism, then there exists $f_0 \in \mathcal{M} (N)$ such that $φ(g) = f_0 g f_0^{-1}$ for all $g \in \mathcal{G}$. We deduce that the abstract commensurator of $\mathcal{M} (N)$ coincides with $\mathcal{M} (N)$.

Group localization and two problems of Levine

A. K. Bousfield's $H\mathbb Z$-localization of groups inverts homologically two-connected homomorphisms of groups. J. P. Levine's algebraic closure of groups inverts homomorphisms between finitely generated and finitely presented groups which are homologically two-connected and for which the image normally generates. We resolve an old problem concerning Bousfield $H\mathbb Z$-localization of groups, and answer two questions of Levine regarding algebraic closure of groups. In particular, we show that the kernel of the natural homomorphism from a group $G$ to it's Bousfield $H\mathbb Z$-localiza…

Completely reducible SL(2)-homomorphisms

Let K be any field, and let G be a semisimple group over K. Suppose the characteristic of K is positive and is very good for G. We describe all group scheme homomorphisms phi:SL(2) --> G whose image is geometrically G-completely reducible -- or G-cr -- in the sense of Serre; the description resembles that of irreducible modules given by Steinberg's tensor product theorem. In case K is algebraically closed and G is simple, the result proved here was previously obtained by Liebeck and Seitz using different methods. A recent result shows the Lie algebra of the image of phi to be geometrically G-c…

On abstract representations of the groups of rational points of algebraic groups in positive characteristic

We analyze the structure of a large class of connected algebraic rings over an algebraically closed field of positive characteristic using Greenberg's perfectization functor. We then give applications to rigidity problems for representations of Chevalley groups, recovering, in particular, a rigidity theorem of Seitz.

Bockstein homomorphisms for Hochschild cohomology of group algebras and of block algebras of finite groups

We give an explicit approach for Bockstein homomorphisms of the Hochschild cohomology of a group algebra and of a block algebra of a finite group and we show some properties. To give explicit definitions for these maps we use an additive decomposition and a Product Formula for the Hochschild cohomology of group algebras given by Siegel and Witherspoon in 1999. For k an algebraically closed field of characteristic p and G a finite group we prove an additive decomposition and a Product Formula for the cohomology algebra of a defect group of a block ideal of kG with coefficients in the source alg…

On a Generalization of the Notion of Semidirect Product of Groups

We introduce an external version of the internal r-fold semidirect product of groups (SDP) of Carrasco and Cegarra. Just as for the classical external SDP, certain algebraic data are required to guarantee associativity of the construction. We give an algorithmic procedure for computing axioms characterizing these data. Additionally, we give criteria for determining when a family of homomorphisms from the factors of an SDP into a monoid or group assemble into a homomorphism on the entire SDP. These tools will be used elsewhere to give explicit algebraic axioms for hypercrossed complexes, which …

On injective homomorphisms for pure braid groups, and associated Lie algebras

The question of whether a representation of Artin's pure braid group is faithful is translated to certain properties of the Lie algebra arising from the descending central series of the pure braid group, and thus the Vassiliev invariants of pure braids via work of T. Kohno \cite{kohno1,kohno2}. The main result is a Lie algebraic condition which guarantees that a homomorphism out of the classical pure braid group is faithful. However, it is unclear whether the methods here can be applied to any open cases such as the Gassner representation.

An algebraic characterization of $k$--colorability

We characterize $k$--colorability of a simplicial graph via the intrinsic algebraic structure of the associated right-angled Artin group. As a consequence, we show that a certain problem about the existence of homomorphisms from right-angled Artin groups to products of free groups is NP--complete.

Quillen Suslin theory for algebraic fundamental group

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 the end of the paper, we also propose some problems related to the algebraic fundamental group of some particular type of rings.

Abstract involutions of algebraic groups and of Kac-Moody groups

Based on the second author's thesis in this article we provide a uniform treatment of abstract involutions of algebraic groups and of Kac-Moody groups using twin buildings, RGD systems, and twisted involutions of Coxeter groups. Notably we simultaneously generalize the double coset decompositions established by Springer and by Helminck-Wang for algebraic groups and by Kac-Wang for certain Kac-Moody groups, we analyze the filtration studied by Devillers-Muhlherr in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings rai…

Operator-algebraic superrigidity for $SL_n(\mathbb Z),n\geq 3$

For $n\geq 3,$ let $Γ=SL_n(\mathbb Z).$ We prove the following superridigity result for $Γ$ in the context of operator algebras. Let $L(Γ)$ be the von Neumann algebra generated by the left regular representation of $Γ.$ Let $M$ be a finite factor and let $U(M)$ be its unitary group. Let $π: Γ\to U(M)$ be a group homomorphism such that $π(Γ)''=M.$ Then \begin{itemize} \item[(i)] either $M$ is finite dimensional, or \item [(ii)] there exists a subgroup of finite index $Λ$ of $Γ$ such that $π|_Λ$ extends to a homomorphism $U(L(Λ))\to U(M).$ \end{itemize} The result is deduced from a complete desc…

直角 Artin 群函子作为范畴嵌入

The Right Angled Artin Group Functor as a Categorical Embedding

本文证明可通过纯代数方式在群中刻画直角 Artin 群，并在群同态中刻画图同态，进而给出从 raag 恢复原图的新算法。