Emmanuel Breuillard, Ben Green
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.
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On the orbits of a finite solvable primitive linear group
该研究聚焦有限可解本原线性群的轨道数量问题，对Seager的相关结论进行了强化，推进了群论领域该方向的研究。
Separability of Solvable Subgroups in Linear Groups
设G是特征为0的域上的非线性生成的线性群.设G的每一个可解子群都是循环的。然后，声称任何可解子群G是可分的。这一点在G=SL_n（Z）中得到了证明。然而，第4节中的主要定理的证明是不完整的。
We present an algorithm to convert a word of length $n$ in the standard generators of the solvable Baumslag-Solitar group $BS(1,p)$ into a geodesic word, which runs in linear time and $O(n\log n)$ space on a random access machine.
Non-linear, solvable, residually $p$ groups
本文构造了有限生成非线性可解剩余2群的第一个例子，填补了从剩余有限群到剩余p群这一研究方向的空白。
This work provides a method(an algorithm) for solving the solvable unary algebraic equation $f(x)=0$ ($f(x)\in\mathbb{Q}[x]$) of arbitrary degree and obtaining the exact radical roots. This method requires that we know the Galois group as the permutation group of the roots of $f(x)$ and the approximate roots with sufficient precision beforehand. Of course, the approximate roots are not necessary but can help reduce the quantity of computation. The algorithm complexity is approximately proportional to the 4th power of the size of the Galois group of $f(x)$. The whole algorithm doesn't need to d…
On the Orbits of Solvable Linear Groups
设$G$是作用在有限向量空间$V$上的可解线性群，并假定$(|G|,|V|)=1$。在本文中，我们发现$x,y\in V$，使$C_G(x)\cap C_G(y)=1$。特别是，这回答了我的一个问题。M.艾萨克我们完善了S. Dolphi，A. Seress和T. R.狼
Infra-Solvmanifolds and Rigidity of Subgroups in Solvable Linear Algebraic Groups
给出了具有同构基本群的紧亚解流形是光滑亚纯的一个新的证明。更一般地说，我们证明了刚性的流形上使用仿射作用的虚拟多环群可解李群。我们的结果是由可解线性代数群中子群的刚性性质得到的。
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。
Let $G$ be a finite group, and let $V$ be a completely reducible faithful $G$-module. By a result of Glauberman it has been known for a long time that if $G$ is nilpotent of class 2, then $|G| < |V|$. In this paper we generalize this result as follows. Assuming $G$ to be solvable, we show that the order of the maximal class 2 quotient of $G$ is strictly bounded above by $|V|$.
We develop a practical algorithm to decide whether a finitely generated subgroup of a solvable algebraic group $G$ is arithmetic. This incorporates a procedure to compute a generating set of an arithmetic subgroup of $G$. We also provide a simple new algorithm for integrality testing of finitely generated solvable-by-finite linear groups over the rational field. The algorithms have been implemented in {\sc Magma}.
Let $V$ be a finite vector space over a finite field of order $q$ and of characteristic $p$. Let $G\leq GL(V)$ be a $p$-solvable completely reducible linear group. Then there exists a base for $G$ on $V$ of size at most $2$ unless $q \leq 4$ in which case there exists a base of size at most $3$. The first statement extends a recent result of Halasi and Podoski and the second statement generalizes a theorem of Seress. An extension of a theorem of Pálfy and Wolf is also given.
From Thompson to Baer-Suzuki: a sharp characterization of the solvable radical
我们证明了素数阶元$g$属于有限群（或更一般地说，线性群）的可解根$R(G)$当且仅当对每个$x\in G$，由$g, xgx^{-1}$生成的子群是可解的.这个定理意味着有限（或线性）群$G$是可解的当且仅当在$G$的每个共轭类中每两个元素生成一个可解子群。
Rational cohomology and Zariski dense subgroups of solvable linear algebraic groups
本文研究了可解线性代数群中扎里斯基稠密子群的上同调性质，证明了在特定条件下，群的上同调环与子群的上同调环存在同构，并探讨了有限生成可解群的上同调表示。
In this work we study the structure of finitely generated groups for which a space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse direction to Kleiner's theorem). We prove that this is indeed the case for solvable groups. The investigation is partly motivated by Kleiner's proof for Gromov's theorem on groups of polynomial growth.
The solvable Farb growth of a group quantifies how well-approximated the group is by its finite solvable quotients. In this note we present a new characterization of polycyclic groups which are virtually nilpotent. That is, we show that a group has solvable Farb growth which is at most polynomial in $\log(n)$ if and only if the group is polycyclic and virtually nilpotent. We also give new results concerning approximating oriented surface groups by nilpotent quotients. As a consequence of this, we prove that a natural number $C$ exists so that any nontrivial element of the $Ck$th term of the lo…
Solvability in Groups of Piecewise-linear Homeomorphisms of the Unit Interval
本文研究了单位区间上具有多个斜率突变的分段线性保向同胚群PLo（I）的子群以及Thompson群F的子群。我们找到了确定任何这样的群的导出长度的几何准则，并利用这个准则对PLo（I）和F的可解和不可解子群进行了分类。设H是PL_o（I）或F的子群。我们发现，H是可解的当且仅当H同构于一个良好描述的群类R中的一个群。我们还发现H是不可解的当且仅当我们可以嵌入一个特定的可解群W到H中。我们通过寻找弱几何准则来加强非可解性分类，在该准则下，我们可以将其他群（都包含W）嵌入到PLo（I）或F的非可解子群中。
Base sizes for finite linear groups with solvable stabilisers
本文针对点稳定子可解、可解根基平凡的传递置换群的基大小猜想，证明了基柱同构于PSL_n(q)的殆单群情形下该猜想的强形式，其余典型群将在后续论文处理。
Non-solvable groups whose non-linear character degrees have the same number of different prime divisors
本文研究非可解群的非线性特征度数具有相同不同素因子数的性质，证明此类群在可解因子作用下仅限于特定几类群，并验证Huppert的ρ-σ猜想。
We design new deterministic and randomized algorithms for computational problems in free solvable groups. In particular, we prove that the word problem and the power problem can be solved in quasi-linear time and the conjugacy problem can be solved in quasi-quartic time by Monte Carlo type algorithms.
Let $V$ be a faithful $G$-module for a finite group $G$ and let $p$ be a prime dividing $|G|$. An orbit $v^G$ for the action of $G$ on $V$ is $p$-regular if $|v^G|_p=|G:\bC_G(v)|_p=|G|_p$. Zhang asks the following question in \cite{Zhang}. Assume that a finite solvable group $G$ acts faithfully and irreducibly on a vector space $V$ over a finite field $\FF$. If $G$ has a $p$-regular orbit for every prime $p$ dividing $|G|$, is it true that $G$ will have a regular orbit on $V$? In \cite{LuCao}, Lü and Cao construct an example showing that the answer to this question is no, however the example i…
We consider the Noether's problem on the noncommutative real rational functions invariant under the linear action of a finite group. For abelian groups the invariant skew-fields are always rational. We show that for a solvable group the invariant skew-field is finitely generated. The skew-field invariant under a linear action of a solvable group is rational if the action is well-behaved -- given by a so-called complete representation. We determine the groups that admit such representations and call them totally psuedo-unramified. In the second part we study the reach of totally psuedo-unramifi…
Given a finitely generated group $G$, we are interested in common geometric properties of all graphs of faithful actions of $G$. In this article we focus on their growth. We say that a group $G$ has a Schreier growth gap $f(n)$ if every faithful $G$-set $X$ satisfies $\mathrm{vol}_{G, X}(n)\succcurlyeq f(n)$, where $\mathrm{vol}_{G, X}(n)$ is the growth of the action of $G$ on $X$. Here we study Schreier growth gaps for finitely generated solvable groups. We prove that if a metabelian group $G$ is either finitely presented or torsion-free, then $G$ has a Schreier growth gap $n^2$, provided $G$…
We extend classical density theorems of Borel and Dani--Shalom on lattices in semisimple, respectively solvable algebraic groups over local fields to approximate lattices. Our proofs are based on the observation that Zariski closures of approximate subgroups are close to algebraic subgroups. Our main tools are stationary joinings between the hull dynamical systems of discrete approximate subgroups and their Zariski closures.
A Minimal Non-solvable Group of Homeomorphisms
设$PL_0(I)$表示单位区间上在复合运算下允许有100多个斜率突变的保向分段线性同胚群。我们找到了一个非可解群$W$，并证明了$W$嵌入到$PL_0(I)$的每个非可解子群中.我们发现温和的条件下，其他不可解的子群（$B$，$(\wr\Z\wr)^{\infty}$，$(\Z\wr)^{\infty}$，和$^{\infty}(\wr\Z)$）嵌入子群的$PL_0(I)$。证明了$PL_0(I)$的所有可解子群嵌入$PL_0(I)$的所有不可解子群中.如果我们用任意广义R代替$PL_0(I)$，这些结果仍然适用。Thompson组$F_n$。
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 …
Kernels of minimal characters of solvable groups
本文研究有限可解群中极小次数非线性不可约特征标，证明当其次数为奇数时，相应特征标核所得商群具有幂零-by-阿贝尔结构。
A set $B$ is a basis for a vector space $V$ if every element of $V$ can be uniquely written as a linear combination of the elements of $B$. There is a similar definition of a basis for a finite group. We show that certain semidirect products of finite groups---including all semidirect products of finite abelian groups---have bases; any group of order $m$ or $2m$ for odd, cube-free $m$ has a basis; and the quaternions do not have a basis.
Let $Q$ be a $p$-subgroup of a finite $p$-solvable group $G$, where $p$ is a prime, and suppose that $δ$ is a linear character of $Q$ with the property that $δ(x)=δ(y)$ whenever $x,y\in Q$ are conjugate in $G$. In this situation, we show that restriction to $p$-regular elements defines a canonical bijection from the set of those irreducible ordinary characters of $G$ with Navarro vertex $(Q,δ)$ onto the set of irreducible Brauer characters of $G$ with Green vertex $Q$. Also, we use this correspondence to examine the behavior of lifts of Brauer characters with respect to normal subgroups.
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