The point-pushing subgroup P(S) of the mapping class group Mod(S) of a surface with marked point is an embedding of π_1(S) given by pushing the marked point around loops. We prove that for g>=3, the subgroup P(S) is the unique normal, genus g surface subgroup of Mod(S). As a corollary to this uniqueness result, we give a new proof that Out(Mod^{\pm}(S))=1, where Out denotes the outer automorphism group; a proof which does not use automorphisms of complexes of curves. Ingredients in our proof of this characterization theorem include combinatorial group theory, representation theory, the Johnson theory of the Torelli group, surface topology, and the theory of Lie algebras.
This paper is a contribution to the study of the subgroup structure of exceptional algebraic groups over algebraically closed fields of arbitrary characteristic. Following Serre, a closed subgroup of a semisimple algebraic group $G$ is called irreducible if it lies in no proper parabolic subgroup of $G$. In this paper we complete the classification of irreducible connected subgroups of exceptional algebraic groups, providing an explicit set of representatives for the conjugacy classes of such subgroups. Many consequences of this classification are also given. These include results concerning t…
It was shown in Part I that there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski-dense. Here we show that the same is true for as long as the transcendence degree of the field is at least $1$ in characteristic zero and at least $2$ in positive characteristic. We also consider related questions for surface groups.
The additivity with respect to exact sequences is notoriously a fundamental property of the algebraic entropy of group endomorphisms. It was proved for abelian groups by deeply exploiting their structure. On the other hand, a solvable counterexample was recently found, showing that it does not hold in general. Nevertheless, we give a rather short proof of the additivity of the algebraic entropy for locally finite groups that are either quasihamiltonian or FC-groups.
Let $G$ be a simple algebraic group over an algebraically closed field $k$, where $\mathrm{char}\, k$ is either 0 or a good prime for $G$. We consider the modality $\mathrm{mod}(B : \mathfrak u)$ of the action of a Borel subgroup $B$ of $G$ on the Lie algebra $\mathfrak u$ of the unipotent radical of $B$, and report on computer calculations used to show that $\mathrm{mod}(B:\mathfrak u) = 20$, when $G$ is of type $\mathrm E_8$. This completes the determination of the values for $\mathrm{mod}(B:\mathfrak u)$ for $G$ of exceptional type.
We prove that various subgroups of the mapping class group $Mod(Σ)$ of a surface $Σ$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the "point-pushing" and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute lower bounds on distortion via representation theory and an extension of Johnson theory to arbitrary subgroups of $H_1(Σ;\mathbb{Z})$.
Let $k$ be a nonperfect separably closed field. Let $G$ be a connected reductive algebraic group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In particular, we present the first example of a connected nonabelian $k$-subgroup $H$ of $G$ that is $G$-completely reducible but not $G$-completely reducible over $k$, and the first example of a connected nonabelian $k$-subgroup $H'$ of $G$ that is $G$-completely reducible over $k$ but not $G$-completely reducible. This is new: all previously known such examples are for finite (or non-…
We show that, in a weakly regular $p$-adic Lie group $G$, the subgroup $G_u$ spanned by the one-parameter subgroups of $G$ admits a Levi decomposition. As a consequence, there exists a regular open subgroup of $G$ which contains $G_u$.
The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of $G$, then the subgroup is called non-generic. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic. A finite subgroup is called Lie primitive if it lies in no proper subgroup of positive dimension. We prove here that many non-generic subgroup types, including the alternating and symmetric…
We investigate Levi subgroups of a connected reductive algebraic group G, over a ground field K. We parametrize their conjugacy classes in terms of sets of simple roots and we prove that two Levi K-subgroups of G are rationally conjugate if and only if they are geometrically conjugate. These results are generalized to arbitrary connected linear algebraic K-groups. In that setting the appropriate analogue of a Levi subgroup is derived from the notion of a pseudo-parabolic subgroup.
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…
Let $k$ be a nonperfect separably closed field. Let $G$ be a (possibly non-connected) reductive group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In our previous work, we constructed examples of subgroups $H$ of $G$ that are $G$-completely reducible but not $G$-completely reducible over $k$ (and vice versa). In this paper, we give a theoretical underpinning of those constructions. To illustrate our result, we present a new such example in a non-connected reductive group of type $D_4$ in characteristic $2$. Then using Geometri…
Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial finite dimensional irreducible rational $KG$-module. We say that $(G,H,V)$ is an irreducible triple if $V$ is irreducible as a $KH$-module. Determining these triples is a fundamental problem in the representation theory of algebraic groups, which arises naturally in the study of the subgroup structure of classical groups. In the 1980s, Seitz and Testerman extended earlier work of Dynkin on connected subgroups…
Let $G$ be a semisimple linear algebraic group over a field $k$ and let $G^+(k)$ be the subgroup generated by the subgroups $R_u(Q)(k)$, where $Q$ ranges over all the minimal $k$-parabolic subgroups $Q$ of $G$. We prove that if $G^+(k)$ is bounded then it is uniformly bounded. Under extra assumptions we get explicit bounds for $Δ(G^+(k))$: we prove that if $k$ is algebraically closed then $Δ(G^+(k))\leq 4\, {\rm rank}\,G$, and if $G$ is split over $k$ then $Δ(G^+(k))\leq 28\, {\rm rank}\,G$. We deduce some analogous results for real and complex semisimple Lie groups.
The purpose of this paper is to link anisotropy properties of an algebraic group together with compactness issues in the topological group of its rational points. We nd equivalent conditions on a smooth ane algebraic group scheme over a non-Archimedean local eld for the associated rational points to admit maximal compact subgroups. We use the structure theory of pseudo-reductive groups provided, whatever the characteristic, by Conrad, Gabber and Prasad. We also investigate thoroughly maximal prop subgroups in the semisimple case, using Bruhat-Tits theory.
Let $G$ be a simple algebraic group over an algebraically closed field. A closed subgroup $H$ of $G$ is called $G$-completely reducible ($G$-cr) if, whenever $H$ is contained in a parabolic subgroup $P$ of $G$, it is contained in a Levi factor of $P$. In this paper we complete the classification of connected $G$-cr subgroups when $G$ has exceptional type, by determining the $L_{0}$-irreducible connected reductive subgroups for each simple classical factor $L_{0}$ of a Levi subgroup of $G$. As an illustration, we determine all reducible, $G$-cr semisimple subgroups when $G$ has type $F_4$ and v…
Given a locally finite leafless tree $ T $, various algebraic groups over local fields might appear as closed subgroups of $ \text{Aut} (T)$. We show that the set of closed cocompact subgroups of $ \text{Aut} (T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $ \text{Aut} (T)$. This is done via a study of the integral Bruhat$-$Tits model of $ \text{SL}_2 $ and $ \text{SU}_3^{L/K} $, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic…
We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.
Let $D$ be a division ring and $K$ a subfield of $D$ which is not necessarily contained in the center $F$ of $D$. In this paper, we study the structure of $D$ under the condition of left algebraicity of certain subsets of $D$ over $K$. Among results, it is proved that if $D^*$ contains a noncentral normal subgroup which is left algebraic over $K$ of bounded degree $d$, then $[D:F]\le d^2$. In case $K=F$, the obtained results show that if either all additive commutators or all multiplicative commutators with respect to a noncentral subnormal subgroup of $D^*$ are algebraic of bounded degree $d$…
The notion of a \emph{$G$-completely reducible} subgroup is important in the study of algebraic groups and their subgroup structure. It generalizes the usual idea of complete reducibility from representation theory: a subgroup $H$ of a general linear group $G= {\rm GL}_n(k)$ is $G$-completely reducible if and only if the inclusion map $i\colon H\rightarrow {\rm GL}_n(k)$ is a completely reducible representation of $H$. In these notes I give an introduction to the theory of complete reducibility and its applications, and explain an approach to the subject using geometric invariant theory.
We study verbally closed subgroups of free solvable groups. A number of results is proved that give sufficient conditions under whose a verbally closed subgroup is turned to be a retract and so algebraically closed of the full group.
