On solvable spherical subgroups of semisimple algebraic groups

Non-vanishing for group $L^p$-cohomology of solvable and semisimple Lie groups

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 Gromov's question 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.

Conjugation of semisimple subgroups over real number fields of bounded degree

Let $G$ be a linear algebraic group over a field $k$ of characteristic 0. We show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over an algebraic closure of $k$ are actually conjugate over a finite field extension of $k$ of degree bounded independently of the subgroups. Moreover, if $k$ is a real number field, we show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over the field of real numbers $\mathbb{R}$ are actually conjugate over a finite real extension of $k$ of degree bounded independently of the subgroups.

Harish-Chandra--Schwartz's algebras associated with discrete subgroups of Semisimple Lie groups

We prove that the Harish-Chandra--Schwartz space associated with a discrete subgroup of a semisimple Lie group is a dense subalgebra of the reduced $C^*$-algebra of the discrete subgroup. Then, we prove that for the reduced $C^*$-norm is controlled by the norm of the Harish-Chandra--Schwartz space. This inequality is weaker than property RD and holds for any discrete group acting isometrically, properly on a Riemannian symmetric space.

Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups

We provide partial results towards a conjectural generalization of a theorem of Lubotzky-Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric inequalities and finiteness properties. As a tool in our proof, we establish polynomial isoperimetric inequalities and finiteness properties for certain solvable groups that appear as subgroups of parabolic groups in semisimple groups, thus generalizing a theorem of Bux. We also develop a precise version of reduction theory for arithmetic groups whose proof is, for …

Strongly dense free subgroups of semisimple algebraic groups

We show that (with one possible exception) 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. As a consequence, we get new generating results for finite simple groups of Lie type and a strengthening of a theorem of Borel related to the Hausdorff-Banach-Tarski paradox. In a sequel to this paper, we use this result to also establish uniform expansion properties for random Cayley graphs over finite simple groups of Lie type.

Strongly dense free subgroups of semisimple algebraic groups II

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.

Strongly solvable spherical subgroups and their combinatorial invariants

A subgroup H of an algebraic group G is said to be strongly solvable if H is contained in a Borel subgroup of G. This paper is devoted to establishing relationships between the following three combinatorial classifications of strongly solvable spherical subgroups in reductive complex algebraic groups: Luna's general classification of arbitrary spherical subgroups restricted to the strongly solvable case, Luna's 1993 classification of strongly solvable wonderful subgroups, and the author's 2011 classification of strongly solvable spherical subgroups. We give a detailed presentation of all the t…

Abelian subgroups and semisimple elements in $2$-modular representations of the symplectic group $Sp_{2n}(2)$

We determine the irreducible 2-modular representations of the symplectic group $G=Sp_{2n}(2)$ whose restriction to every abelian subgroup has a trivial constituent. A similar result is obtained for maximal tori of $G$. There is significant information on the existence of eigenvalue 1 of elements of $G$ in a given irreducible representation of $G$.

无矩条件下半单李群离散子群的Poisson边界

The Poisson boundary of discrete subgroups of semisimple Lie groups without moment conditions

该研究在无需矩条件的前提下，证明了半单李群Zariski稠密离散子群上有限熵随机游走的Poisson边界，就是配击中测度的对应对称空间的Furstenberg边界。

半单李群的离散子群：超越格

Discrete subgroups of semisimple Lie groups, beyond lattices

本文从几何与动力学视角综述了高秩半单李群中超越格的薄离散子群的最新构造与理解进展，并涉及其形变空间与高阶Teichmüller理论。

Subgroup growth of lattices in semisimple Lie groups

We give very precise bounds for the congruence subgroup growth of arithmetic groups. This allows us to determine the subgroup growth of irreducible lattices of semisimple Lie groups. In the most general case our results depend on the Generalized Riemann Hypothesis for number fields but we can state the following unconditional theorem: Let $G$ be a simple Lie group of real rank at least 2, different than $D_4(\bbc)$, and let $Γ$ be any non-uniform lattice of $G$. Let $s_n(Γ)$ denote the number of subgroups of index at most $n$ in $Γ$. Then the limit $\lim\limits_{n\to \infty} \frac{\log s_n(Γ)}…

由半单元素生成的代数群

Algebraic groups generated by semisimple elements

本文刻画了特征零代数闭域上连通线性代数群中由所有半单元素生成的子群。

Locally Solvable Subgroups of PLo(I)

We show that locally solvable subgroups of PLo(I) are countable. Then for each countable ordered set, we construct a locally solvable subgroup of Thompson's Group F. We develop machinery for understanding embeddings from solvable subgroups into solvable subgroups. Finally, we apply this machinery to show the ordered sets used in our construction are invariant under isomorphisms between the groups constructed. Therefore, we effectively distinguish the groups and provide uncountably many non-isomorphic locally solvable, hence elementary amenable, subgroups of Thompson's Group F.

Weak second maximal subgroups in solvable groups

In this paper, we investigate the difference between weak second maximal subgroups and second maximal subgroups. A sufficient and necessary condition is also given to describe such class of groups whose weak second maximal subgroups coincide with its second maximal subgroups(called WSM-groups) under the solvable case. As an application, we show that every non-vanishing element of a solvable WSM-group lies in its Fitting subgroup.

On Levi subgroups and the Levi decomposition for groups definable in o-minimal structures

We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an o-minimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = RS where R is the solvable radical of G. We also prove that any semisimple subalgebra of the Lie algebra of G corresponds to a unique ind-definable semisimple subgroup of G.

Duality between $p$-groups with three characteristic subgroups and semisimple anti-commutative algebras

Let $p$ be an odd prime and let $G$ be a non-abelian finite $p$-group of exponent $p^2$ with three distinct characteristic subgroups, namely $1$, $G^p$, and $G$. The quotient group $G/G^p$ gives rise to an anti-commutative ${\mathbb F}_p$-algebra $L$ such that the action of ${\rm Aut}(L)$ is irreducible on $L$; we call such an algebra IAC. This paper establishes a duality $G\leftrightarrow L$ between such groups and such IAC algebras. We prove that IAC algebras are semisimple and we classify the simple IAC algebras of dimension at most 4 over certain fields. We also give other examples of simp…

Homogeneous spaces of real simple Lie groups with proper actions of non virtually abelian discrete subgroups: a calculational approach

Let G be a simple non-compact linear connected Lie group and H be a closed non-compact semisimple subgroup. We are interested in finding classes of homogeneous spaces G/H admitting proper actions of discrete non virtually abelian subgroups of G. We develop an algorithm for finding such homogeneous spaces. As a testing example we obtain a list of all non-compact homogeneous spaces G/H admitting proper action of a discrete and non virtually abelian subgroup of G in the case when G has rank at most 8, and H is a maximal proper semisimple subgroup.

可解变换群的变种

Solvable Automorphism Groups of Varieties

研究证明，当变种X为准仿射时，由包含单位的不可约族生成的可解子群为代数子群，且连通可解子群导出长度不超过n+1，X同构于仿射空间且Borel子群与Jonquières子群共轭。

On verbally closed subgroups of free solvable groups

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.

On the intersection of solvable Hall subgroups in finite simple exceptional groups of Lie type

Assume that a finite almost simple group with simple socle isomorphic to an exceptional group of Lie type possesses a solvable Hall subgroup. Then there exist four conjugates of the subgroup such that their intersection is trivial.

Locally solvable subgroups of PLo(I) are countable

We show every locally solvable subgroup of PLo(I) is countable. A corollary is that an uncountable wreath product of copies of $\mathbb{Z}$ with itself does not embed into PLo(I).

A Criterion for the Existence of a Solvable $π$-Hall Subgroup in a Finite Group

Let $G$ be a finite group and let $π$ be a set of primes. In this paper, we prove a criterion for the existence of a solvable $π$-Hall subgroup of $G$, precisely, the group $G$ has a solvable $π$-Hall subgroup if, and only if, $G$ has a $\{p,q\}$-Hall subgroup for any pair $p$, $q\inπ$.

Separability of Solvable Subgroups in Linear Groups

Let G be a finitely generated linear group over a field of characteristic 0. Suppose that every solvable subgroup of G is polycyclic. Then the claim is made that any solvable subgroup of G is separable. This is proven for G=SL_n(Z). However, the proof of the main theorem in section 4 is incomplete.

Two-generated verbally closed subgroups of a free solvable group G are retracts of G

We prove that every verbally closed two-generated subgroup of a free solvable group G of a finite rank is a retract of G.

On Maximal Subgroups of a Finite Solvable Group

The following result is received: Let $H$ be a non-normal maximal subgroup of a finite solvable group $G$ and let $q \in π(F(H/\mathrm{Core}_GH))$, then $G$ has a Sylow $q$-subgroup $Q$ such that $N_{G}(Q) \subseteq H$.

Solvable Subgroups of Locally Compact Groups

It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.

Lie rank in groups of finite Morley rank with solvable local subgroups

We prove a general dichotomy theorem for groups of finite Morley rank with solvable local subgroups and of Prüfer p-rank at least 2, leading either to some p-strong embedding, or to the Prüfer p-rank being exactly 2.

从子群集的格到广义可解扩展族的新方法

A New Approach from Lattice of Subgroup Sets to Generalized Solvable Extension Formations

本文提出从子群集格到广义可解扩展族的新方法，通过统一框架和极大子群集确定实现生成形成映射。

Definably semisimple groups interpretable in $p$-adically closed fields

Let $K$ be a $p$-adically closed field and $G$ a group interpretable in $K$. We show that if $G$ is definably semisimple (i.e. $G$ has no definable infinite normal abelian subgroups) then there exists a finite normal subgroup $H$ such that $G/H$ is definably isomorphic to a $K$-linear group. The result remains true in models of $\mathrm{Th}(\mathbb{Q}_p^{an})$.

Subgroups of finitely presented groups with solvable conjugacy problem

We prove that every countable group with solvable power problem embeds into a finitely presented 2-generated group with solvable power and conjugacy problems.

On subnormal subgroups in division rings containing a non-abelian solvable subgroup

Let $D$ be a division ring with center $F$ and $N$ a subnormal subgroup of the multiplicative group $D^*$ of $D$. Assume that $N$ contains a non-abelian solvable subgroup. In this paper, we study the problem on the existence of non-abelian free subgroups in $N$. In particular, we show that if either $N$ is algebraic over $F$ or $F$ is uncountable, then $N$ contains a non-abelian free subgroup.

On solvable subgroups of automorphism groups of right-angled Artin groups

For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph that determines which case holds. We also consider some examples of solvable subgroups, including one that is not virtually nilpotent and is embedded in a non-obvious way.

Intersection of Solvable Hall subgroups in finite groups

In this paper we show that if S is a simple classical group, a group G is contained in inner-diagonal automorphisms of S and contains S, and H is a solvable Hall subgroup of G, then there exists five conjugates of H, whose intersection is trivial.

Intersection of conjugate solvable subgroups in finite classical groups

We consider the following problem stated by Vdovin (2010) in the "Kourovka notebook" (Problem 17.41): Let $H$ be a solvable subgroup of a finite group $G$ that has no nontrivial solvable normal subgroups. Do there always exist five conjugates of $H$ whose intersection is trivial? This problem is closely related to a conjecture by Babai, Goodman and Pyber (1997) about an upper bound for the index of a normal solvable subgroup in a finite group. In particular, a positive answer to Vdovin's problem yields that if $G$ has a solvable subgroup of index $n$, then it has a solvable normal subgroup of …

具有素数幂指数可解子群的群

Groups with a solvable subgroup of prime-power index

本文研究了包含有限素数幂指数可解子群的群的性质，证明了此类群的商群在某些条件下会渐近趋于小。

Intersection of conjugate solvable subgroups in symmetric groups

It is shown that for a solvable subgroup $G$ of an almost simple group $S$ which socle is isomorphic to $A_n$ $ (n\ge5)$ there are $x,y,z,t \in S$ such that $G \cap G^x \cap G^y \cap G^z \cap G^t =1.$

Superrigid subgroups and syndetic hulls in solvable Lie groups

This is an expository paper. It is not difficult to see that every group homomorphism from the additive group Z of integers to the additive group R of real numbers extends to a homomorphism from R to R. We discuss other examples of discrete subgroups D of connected Lie groups G, such that the homomorphisms defined on D can ("virtually") be extended to homomorphisms defined on all of G. For the case where G is solvable, we give a simple proof that D has this property if it is Zariski dense. The key ingredient is a result on the existence of syndetic hulls.

Some results on a question of M. Newman on isomorphic subgroups of solvable groups

In this paper, we focus on a question of M. Newman on isomorphic subgroups of solvable groups. We get a reduction theorem of this question: for each prime q, assume that this question holds for every characteristic q-groups, then this question holds for every finite solvable groups. Using this reduction theorem, we get some partial answers about this question.

The BNS invariants of the generalized solvable Baumslag-Solitar groups and of their finite index subgroups

We compute the Bieri-Neumann-Strebel invariants $Σ^1$ for the generalized solvable Baumslag-Solitar groups $Γ_n$ and their finite index subgroups. Using $Σ^1$, we show that certain finite index subgroups of $Γ_n$ cannot be isomorphic to $Γ_{k}$ for any $k$. In addition, we use the BNS-invariants to give a new proof of property $R_\infty$ for the groups $Γ_n$ and their finite index subgroups.

Groups of finite Morley rank with solvable local subgroups

We lay down the fundations of the theory of groups of finite Morley rank in which local subgroups are solvable and we proceed to the local analysis of these groups. We prove the main Uniqueness Theorem, analogous to the Bender method in finite group theory, and derive its corollaries. We also consider homogeneous cases as well as torsion.

Solvable Subgroup Theorem for simplicial nonpositive curvature

Given a group $G$ with bounded torsion that acts properly on a systolic complex, we show that every solvable subgroup of $G$ is finitely generated and virtually abelian of rank at most $2$. In particular this gives a new proof of the above theorem for systolic groups. The main tools used in the proof are the Product Decomposition Theorem and the Flat Torus Theorem.

Infra-Solvmanifolds and Rigidity of Subgroups in Solvable Linear Algebraic Groups

We give a new proof that compact infra-solvmanifolds with isomorphic fundamental groups are smoothly diffeomorphic. More generally, we prove rigidity results for manifolds which are constructed using affine actions of virtually polycyclic groups on solvable Lie groups. Our results are derived from rigidity properties of subgroups in solvable linear algebraic groups.

实闭域上的半单代数群

Semisimple algebraic groups over real closed fields

本文研究实闭域上半单代数群的性质，推广了半单李群的结果，证明了相关分解定理和Jacobson-Morozov引理，并利用模型论原理进行分析。

可解线性群的Hall子群的大轨道

Large orbits of Hall subgroups of solvable linear groups

本文证明有限可解群中Hall子群的最大特征标次数平方至少为其某正规商群的π部分，刻画了可解线性群中Hall子群的大轨道性质。

Non virtually solvable subgroups of mapping class groups have non virtually solvable representations

Let $Σ$ be a compact orientable surface of finite type with at least one boundary component. Let $Γ\leq \textup{Mod}(Σ)$ be a non virtually solvable subgroup. We answer a question of Lubotzky by showing that there exists a finite dimensional homological representation $ρ$ of $\textup{Mod}(Σ)$ such that $ρ(Γ)$ is not virtually solvable. We then apply results of Lubotzky and Meiri to show that for any random walk on such a group the probability of landing on a power, or on an element with topological entropy $0$ both decrease exponentially in the length of the walk.

Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_{1}}\cdots A^{x_{k}}$ for $A\leq G$ and $x_{1},\ldots ,x_{k}\in G$, where $A$ is nilpotent or solvable. First we exploit the split $BN$-pair structure of finite simple groups of Lie type to give a unified self-contained proof that every such group is a product of four or three unipotent Sylow subgroups. Then we derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group. Finally, using conjugate factorizations of a general finite solvable group by any of its Carter…

具有幂零正规补子群的有限可解群

Finite solvable groups with a nilpotent normal complement subgroup

本文研究有限可解群中非正规无核子群的结构，证明在特定正规化子条件下，该子群存在幂零正规补，且相关构造形成Frobenius群。

Residual finiteness and strict distortion of cyclic subgroups of solvable groups

We provide polynomial lower bounds for residual finiteness of residually finite, finitely generated solvable groups that admit infinite order elements in the Fitting subgroup of strict distortion at least exponential. For this class of solvable groups which include polycyclic groups with a nontrivial exponential radical and the metabelian Baumslag-Solitar groups, we improve the lower bounds found in the literature. Additionally, for the class of residually finite, finitely generated solvable groups of infinite Prüfer rank that satisfy the conditions of our theorem, we provide the first nontriv…

半单李群中的 Dehn 填充

Dehn filling in semisimple Lie groups

将Thurston双曲Dehn填充定理推广至任意秩半单李群，证明扩展几何有限子群的形变保持性质，并给出相对Anosov子群形变保持Anosov的准则，适用于凸射影几何高秩例子。

GL(2, q)中完全可约的无立方因子可解p'-子群的共轭类

Conjugacy classes of completely reducible cube-free solvable p'-subgroups of GL(2, q)

本文针对无立方因子正整数m与不整除m的素数p，确定了一般线性群GL(2, q)（q为p的幂）中阶为m的完全可约可解无立方因子子群的共轭类数量。

具有5个非平凡非自正规子群共轭类的群是可解的

Groups with 5 nontrivial conjugacy classes of non self-normalizing subgroups are solvable

本研究针对Bianchi等人提出的关于$\mathcal{D}_5$（恰有5个非平凡非自正规子群共轭类的有限群族）由导出长度不超过3的可解群组成的猜想，完成验证并证实该猜想成立。

Approximate groups, II: the solvable linear case

We describe the structure of "K-approximate subgroups'' of solvable subgroups of GL_n(C), showing that they have a large nilpotent piece. By combining this with the main result of our recent paper on approximate subgroups of torsion-free nilpotent groups, we show that such approximate subgroups are efficiently controlled by nilpotent progressions.

A Minimal Non-solvable Group of Homeomorphisms

Let $PL_0(I)$ represent the group of orientation-preserving piecewise-linear homeomorphisms of the unit interval which admit finitely many breaks in slope, under the operation of composition. We find a non-solvable group $W$ and show that $W$ embeds in every non-solvable subgroup of $PL_0(I)$. We find mild conditions under which other non-solvable subgroups ($B$, $(\wr\Z\wr)^{\infty}$, $(\Z\wr)^{\infty}$, and $^{\infty}(\wr\Z)$) embed in subgroups of $PL_0(I)$. We show that all solvable subgroups of $PL_0(I)$ embed in all non-solvable subgroups of $PL_0(I)$. These results continue to apply if …

More on a question of M. Newman on isomorphic subgroups of solvable groups

M.Newman has asked if it is the case that whenever H and K are isomorphic subgroups of a finite solvable group G with H maximal, then K is also maximal. This question was considered in a paper of I.M. Isaacs and the second author, where (among other things) the answer was shown to be affirmative if H has an Abelian Sylow 2-subgroup. Here, we show that the answer is affirmative unless the index of H is a power of a prime less than 5 and we obtain further restrictions on the structure of a purported minimal counterexample.