A. Brudnyi
We study the structure of discrete subgroups of the group $G[[r]]$ of complex formal power series under the operation of composition of series. In particular, we prove that every finitely generated fully residually free group is embeddable to $G[[r]]$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, I show that if $p$ is an odd prime, and if $P$ is a finite $p$-group, then there exists an exact sequence of abelian groups $$0\to T(P)\to D(P)\to\lproj{P}\to H^1\big(\apdeux(P),\Z\big)^{(P)},$$ where $D(P)$ is the Dade group of $P$ and $T(P)$ is the subgroup of endo-trivial modules. Here $\lproj{P}$ is the group of sequences of compatible elements in the Dade groups $D\big(N_P(Q)/Q\big)$ for non trivial subgroups $Q$ of $P$. The poset $\apdeux(P)$ is the set of elementary abelian subgroups of rank at least 2 of $P$, ordered by inclusion. The group $H^1\big(\apdeux(P),\Z\big)^{(…
In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show that no big mapping class group satisfies the strong Tits alternative enjoyed by finite-type mapping class groups. We also give examples of big mapping class groups that fail to satisfy even the classical Tits alternative and give a proof that every countable group appears as a sub…
We provide examples of finitely generated infinite covolume subgroups of $PSL(2,R)^r$ with a "big" limit set, e.g. that contains an open subset of the geometric boundary. They are given by the so called semi-arithmetic Fuchsian groups admitting modular embeddings.
In this paper, we study the action of finite subgroups of the mapping class group of a surface on the curve complex. We prove that if the diameter of the almost fixed point set of a finite subgroup H is big enough, then the centralizer of H is infinite.
In the present paper, which is a direct sequel of our paper [12] joint with Roozbeh Hazrat, we prove unrelativised version of the standard commutator formula in the setting of Chevalley groups. Namely, let $Φ$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $I,J$ be two ideals of $R$. We consider subgroups of the Chevalley group $G(Φ,R)$ of type $Φ$ over $R$. The unrelativised elementary subgroup $E(Φ,I)$ of level $I$ is generated (as a group) by the elementary unipotents $x_α(ξ)$, $α\inΦ$, $ξ\in I$, of level $I$. Obviously, in general $E(Φ,I)$ has n…
Let $G$ be a Lie group over a totally disconnected local field and $α$ be an analytic endomorphism of $G$. The contraction group of $α$ ist the set of all $x\in G$ such that $α^n(x)\to e$ as $n\to\infty$. Call sequence $(x_{-n})_{n\geq 0}$ in $G$ an $α$-regressive trajectory for $x\in G$ if $α(x_{-n})=x_{-n+1}$ for all $n\geq 1$ and $x_0=x$. The anti-contraction group of $α$ is the set of all $x\in G$ admitting an $α$-regressive trajectory $(x_{-n})_{n\geq 0}$ such that $x_{-n}\to e$ as $n\to\infty$. The Levi subgroup is the set of all $x\in G$ whose $α$-orbit is relatively compact, and such t…
We construct hyperbolic groups with the following properties: The boundary of the group has big dimension, it is separated by a Cantor set and the group does not split. This shows that Bowditch's theorem that characterizes splittings of hyperbolic groups over 2-ended groups in terms of the boundary can not be extended to splittings over more complicated subgroups.
We study stable commutator length on mapping class groups of certain infinite-type surfaces. In particular, we show that stable commutator length defines a continuous function on the commutator subgroups of such infinite-type mapping class groups. We furthermore show that the commutator subgroups are open and closed subgroups and that the abelianizations are finitely generated in many cases. Our results apply to many popular infinite-type surfaces with locally coarsely bounded mapping class groups.
Let IET be the group of bijections from $\mathopen{[}0,1 \mathclose{[}$ to itself that are continuous outside a finite set, right-continuous and piecewise translations. The abelianization homomorphism $f: \text{IET} \to A$, called SAF-homomorphism, was described by Arnoux-Fathi and Sah. The abelian group $A$ is the second exterior power of the reals over the rationals. For every subgroup $Γ$ of $\mathbb{R/Z}$ we define $\text{IET}(Γ)$ as the subgroup of $\text{IET}$ consisting of all elements $f$ such that $f$ is continuous outside $Γ$. Let $\tildeΓ$ be the preimage of $Γ$ in $\mathbb{R}$. We …
Let S be a surface and let Mod(S,K) be the mapping class group of S permuting a Cantor subset K of S. We prove two structure theorems for normal subgroups of Mod(S,K). (Purity:) if S has finite type, every normal subgroup of Mod(S,K) either contains the kernel of the forgetful map to the mapping class group of S, or it is `pure', i.e. it fixes the Cantor set pointwise. (Inertia:) for any n element subset Q of the Cantor set, there is a forgetful map from the pure subgroup PMod(S,K) of Mod(S,K) to the mapping class group of (S,Q) fixing Q pointwise. If N is a normal subgroup of Mod(S,K) contain…
We initiate the study of subgroups $H$ of the general linear group $GL_{\binom{n}{m}}(R)$ over a commutative ring $R$ that contain the $m$-th exterior power of an elementary group $\bigwedge^mE_n(R)$. Each such group $H$ corresponds to a uniquely defined level $(A_0,\dots,A_{m-1})$, where $A_0,\dots,A_{m-1}$ are ideals of $R$ with certain relations. In the crucial case of the exterior squares, we state the subgroup lattice to be standard. In other words, for $\bigwedge^2E_n(R)$ all intermediate subgroups $H$ are parametrized by a single ideal of the ring $R$. Moreover, we characterize $\bigwed…
A new construction of subgroups of big mapping class groups
本文提出了一种新的方法，通过位移映射和多推映射构造大纯映射类群的子群，这些子群不依赖于已知技术，且不通过表面嵌入诱导。
We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these `big' mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.
It is shown that the big free group (the set of countably-long words over a countable alphabet) is almost free, in the sense that any function from the alphabet to a compact topological group factors through a homomorphism. This statement is in fact a simple corollary of the more general result proven below on the extendability of homomorphisms from subgroups (of a certain kind) of the big free group to a compact topological group.
In this note, we prove that the compactly supported mapping class group of a surface containing a genus $3$ subsurface has no realization as a subgroup of the homeomorphism group. We also prove that for certain surfaces with order $6$ symmetries, their mapping class groups have no realization as a subgroup of the homeomorphism group. Examples of such surfaces include the plane minus a Cantor set and the sphere minus a Cantor set.
Let $H$ be a group and $E$ a set such that $H \subseteq E$. We shall describe and classify up to an isomorphism of groups that stabilizes $H$ the set of all group structures that can be defined on $E$ such that $H$ is a subgroup of $E$. A general product, which we call the unified product, is constructed such that both the crossed product and the bicrossed product of two groups are special cases of it. It is associated to $H$ and to a system $\bigl((S, 1_S,\ast), \triangleleft, \, \triangleright, \, f \bigl)$ called a group extending structure and we denote it by $H \ltimes S$. There exists a …
We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the loop graphs introduced by Bavard-Walker. We classify all of the mapping classes in these actions which are loxodromic with a WWPD action on the corresponding loop graph. The WWPD property is a weakening of Bestvina-Fujiwara's weak proper discontinuity and is useful for constructing non-trivial quasimorphisms. We use this classification to give a sufficient criterion for subgroups of big mapping class groups to have infinite-dimensional second bounded cohomology and use this criterion to giv…
Let $K$ be a field, and let $\Aut \,K^2$ be the group of polynomial automorphisms of $K^2$. We investigate which subgroups are linear or not. In characteristic zero, there are small nonlinear subgroups and some big linear subgroups. When $K$ has finite characteristic, the whole group $\Aut\,K^2$ is linear whenever $K$ is finite, and nonlinear otherwise.
Given an imaginary quadratic extension $K$ of $\mathbb Q$, we give a classification of the maximal nonelementary subgroups of the Picard modular group $\operatorname{PSU}_{1,2}(\mathcal O_K)$ preserving a complex geodesic in the complex hyperbolic plane $\mathbb H^2_\mathbb C$. Complementing work of Holzapfel, Chinburg-Stover and Möller-Toledo, we show that these maximal $\mathbb C$-Fuchsian subgroups are arithmetic, arising from a quaternion algebra $\Big(\!\begin{array}{c} D\,,D_K\\\hline\mathbb QQ\end{array} \!\Big)$ for some explicit $D\in\mathbb N-\{0\}$ and $D_K$ the discriminant of $K$.…
A subgroup $G$ of a product $\prod\limits_{i\in\mathbb{N}}G_i$ is \emph{rectangular} if there are subgroups $H_i$ of $G_i$ such that $G=\prod\limits_{i\in\mathbb{N}}H_i$. We say that $G$ is \emph{weakly rectangular} if there are finite subsets $F_i\subseteq \mathbb{N}$ and subgroups $H_i$ of $\bigoplus\limits_{j\in F_i} G_j$ that satisfy $G=\prod\limits_{i\in\mathbb{N}}H_i$. %We say that $G$ is a \emph{subdirect product} of the family $\{G_i\}_{i\in I}$ if $G$ is weakly rectangular and %$G\cap\bigoplus\limits_{i\in I} G_i=\bigoplus\limits_{i\in\mathbb{N}}H_i$. In this paper we discuss when a c…
For every non-nilpotent finite group $G$, there exists at least one proper subgroup $M$ such that $G$ is the setwise product of a finite number of conjugates of $M$. We define $γ_{\text{cp}}\left( G\right) $ to be the smallest number $k$ such that $G$ is a product, in some order, of $k$ pairwise conjugated proper subgroups of $G$. We prove that if $G$ is non-solvable then $γ_{\text{cp}}\left( G\right) \leq36$ while if $G$ is solvable then $γ_{\text{cp}}\left( G\right) $ can attain any integer value bigger than $2$, while, on the other hand, $γ_{\text{cp}}\left( G\right) \leq4\log_{2}\left\vert…
Equivariant formality and the cohomology of subgroups of right-angled Coxeter groups
本文研究了右角Coxeter群的coabelian子群的分类空间模型，通过实moment-angle复形的同调轨道空间构造，并利用Borel等效上同调理论分析其上同调结构，提出图论判据判断coabelian子群的上同调自由性。
Let $X$ be a surface, possibly with boundary. Suppose it has infinite genus or infinitely many punctures, or a closed subset which is a disk with a Cantor set removed from its interior. For example, $X$ could be any surface of infinite type with only finitely many boundary components. We prove that the mapping class group of $X$ does not satisfy the Tits Alternative. That is, Map$(X)$ contains a finitely generated subgroup that is not virtually solvable and contains no nonabelian free group.
Finitely generated subgroups of free groups as formal languages and their cogrowth
本文研究自由群有限生成子群的正则语言，通过Schreier图核构造最小自动机，刻画不可约情形并给出计算共增长级数与熵的有效方法。
The number of Sylow subgroups and a generalization of Mersenne primes
本文证明，若存在有限群具有mp+1个Sylow p-子群且p足够大，则mp+1为素数，改进了M. Hall的定理并部分回答了Brauer问题26，证明使用了解析数论技术。
Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of characteristic $p>0$, and assume $p$ is good for $G$. Let $P$ be a parabolic subgroup with unipotent radical $U$. For $r \ge 1$, denote by $\mathbb{G}_{a(r)}$ the $r$-th Frobenius kernel of $\mathbb{G}_a$. We prove that if the nilpotence class of $U$ is less than $p$, then any embedding of $\mathbb{G}_{a(r)}$ in $U$ lies inside a one-parameter subgroup of $U$, and there is a canonical way in which to choose such a subgroup. Applying this result, we prove that if $p$ is at least as big as the Coxeter number of $G$,…
On the pseudovariety of groups $\mathbf{U} = \displaystyle\bigvee_{p \in \mathbb{P}} {\bf Ab}(p) \ast {\bf Ab}(p-1)$
本文引入有限群伪簇 $\mathbf{U}$，证明其由具有初等阿贝尔导出子群和阿贝尔 Sylow 子群的有限超可解群组成且可判定，并研究了自由群的相关拓扑可判定性及由其生成的亚阿贝尔群簇。
We calculate the derived series of the Riordan group. To do that, we study a nested sequence of its subgroups, herein denoted by $\mathcal G_k$. By means of this sequence, we first obtain the n-th commutator subgroup of the Associated subgroup. This fact allows us to get some related results about certain groups of formal power series and to complete the proof of our main goal, Theorem 1 in this paper.
Property $P_{\text{naive}}$ for non-orientable big Mapping Class Groups
本文提出可定向大映射类群子群满足性质 $P_{\ ext{naive}}$ 的判据，并借助定向二重复盖推广至不可定向无限型曲面映射类群。
Let $G$ be a finite group and $σ=\{σ_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\bigcup_{i\in I}σ_{i}$ and $σ_{i}\cap σ_{j}=\emptyset$ for all $i\neq j$. A chief factor $H/K$ of $G$ is said to be $σ$-central in $G$, if the semidirect product $(H/K)\rtimes(G/C_G(H/K))$ is a $σ_i$-group for some $i\in I$. The group $G$ is said to be $σ$-nilpotent if either $G=1$ or every chief factor of $G$ is $σ$-central. In this paper, we study the properties of a finite group $G=AB$, factorized by two $σ$-nilpotent subgroups $A$ and $B$, and also generalize some k…
It is known that the Kurosh Subgroup Theorem does not hold for pro-$p$ groups of big cardinality. However, a subgroup $G$ of a free pro-$p$ product is projective relative to the Kurosh family of subgroups. In this paper we prove the converse of this fact.
This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let $a_1G_1,...,a_kG_k$ be left cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ covers each element of $G$ at least $m$ times but none of its proper subsystems does. We show that if $G$ is cyclic, or $G$ is finite and $G_1,...,G_k$ are normal Hall subgroups of $G$, then $k\geq m+f([G:\bigcap_{i=1}^kG_i])$, where $f(\prod_{t=1}^r p_t^{α_t})=\sum_{t=1}^rα_t(p_t-1)$ if $p_1,...,p_r$ are distinct primes and $α_1,...,α_r$ are nonnegative integers. When all the $a_i$ are the identity element of $G$ and …
Consider a presentation $\mathcal{P}=<{\bf x}\mid{\bf \bigcup_{i=1}^n r_i}>$. Let ${\bf R_i}$ be the normal closure of the set ${\bf r_i}$ in the free group ${\bf F}$ with basis ${\bf x}$, $\mathcal{P}_i=<{\bf x}\mid{\bf r_i}>$, ${\bf N_i} = \prod_{j\neq i}{\bf R_j}$. In the present article, using geometric techniques of pictures, generators for $\frac{{\bf R_i}\cap {\bf N_i}}{[{\bf R_i}, {\bf N_i}]}$, $i=1,...,n$, are obtained from a set of generators over $\{\mathcal{P}_i\mid i=1,..., n\}$ for $π_2(\mathcal{P})$. As a corollary, we get a sufficient condition for the family $\{{\bf R_1},...,{…
On compact uniformly recurrent subgroups
本文研究 Chabauty 空间中的极小不变闭集，证明由紧集组成的此类集合的并具有紧闭包，并据此推出局部紧群的紧一致循环子群必含于紧正规子群。
Raikov Remainders of Quotients by Raikov-Complete Almost Metrizable Normal Subgroups
本文研究拓扑群商群的Raikov余项性质，刻画了商同态延拓下余项等式成立的充要条件，证明Raikov完备几乎可度量化闭正规子群满足该条件，得出伪紧性可传递的结论。
A sieve formula for chains of $p$-subgroups
本文针对有限群和素数建立筛法公式，以同余式计数特定阶p-子群链，推广Sylow-Frobenius定理，方法基于Wielandt方法。
We propose a criterion for preserving the regularity of a formal language representation when passing from groups to subgroups. We use this criterion to show that the regularity of a positive cone language in a left-orderable group passes to its finite index subgroups, and to show that there exists no left order on a finitely generated acylindrically hyperbolic group such that the corresponding positive cone is represented by a quasi-geodesic regular language. We also answer one of Navas' questions by giving an example of an infinite family of groups which admit a positive cone that is generat…
For any surface $Σ$ of infinite topological type, we study the Torelli subgroup ${\mathcal I}(Σ)$ of the mapping class group ${\rm MCG}(Σ)$, whose elements are those mapping classes that act trivially on the homology of $Σ$. Our first result asserts that ${\mathcal I}(Σ)$ is topologically generated by the subgroup of ${\rm MCG}(Σ)$ consisting of those elements in the Torelli group which have compact support. In particular, using results of Birman, Powell, and Putman we deduce that ${\mathcal I}(Σ)$ is topologically generated by separating twists and bounding pair maps. Next, we prove the abstr…
In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l at least 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Sp(n, l). This result is for instance useful for proving "big image" results for symplectic Galois representations.
One of the fundamental questions in current field theory, related to Grothendieck's conjecture of birational anabelian geometry, is the investigation of the precise relationship between the Galois theory of fields and the structure of the fields themselves. In this paper we initiate the classification of additive properties of multiplicative subgroups of fields containing all squares, using pro-2-Galois groups of nilpotency class at most 2, and of exponent at most 4. This work extends some powerful methods and techniques from formally real fields to general fields of characteristic not 2.
Let $\mathscr{F}$ be a formation and $G$ a finite group. The weak norm of a subgroup $H$ in $G$ with respect to $\mathscr{F}$ is defined by $N_{\mathscr{F}}(G,H)=\underset{T\leq H}{\bigcap}N_G(T^{\mathscr{F}})$. In particular, $N_{\mathscr{F}}(G)=N_{\mathscr{F}}(G,G)$. Let $N^i_{\mathscr{F}}(G)$,$i\geq 1$, be a upper series of $G$ by setting $N^0_{\mathscr{F}}(G)=1$, $N^{i+1}_{\mathscr{F}}(G)/N^i_{\mathscr{F}}(G)=N_{\mathscr{F}}(G/N^i_{\mathscr{F}}(G))$ and denoted by $N^{\infty}_{\mathscr{F}}(G)$ the terminal term of the series. In this paper, for the case $\mathscr{F}\in\{\mathscr{U}_p,\math…
Intersection-saturated groups without free subgroups
本研究提出交饱和群的新判据，据此找到包括Thompson群、Grigorchuk群在内的新实例，证明了无交换自由子群的有限表现交饱和群存在，解决了Delgado等人的问题。
Let $I$ be an infinite set, $\{G_i:i\in I\}$ be a family of (topological) groups and $G=\prod_{i\in I} G_i$ be its direct product. For $J\subseteq I$, $p_{J}: G\to \prod_{j\in J} G_j$ denotes the projection. We say that a subgroup $H$ of $G$ is: (i) \emph{uniformly controllable} in $G$ provided that for every finite set $J\subseteq I$ there exists a finite set $K\subseteq I$ such that $p_{J}(H)=p_{J}(H\cap\bigoplus_{i\in K} G_i)$; (ii) \emph{controllable} in $G$ provided that $p_{J}(H)=p_{J}(H\cap\bigoplus_{i\in I} G_i)$ for every finite set $J\subseteq I$; (iii) \emph{weakly controllable} in …
Let $G$ be a finite group and $σ=\{σ_{i} | i\in I\}$ some partition of the set of all primes $\Bbb{P}$, that is, $σ=\{σ_{i} | i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} σ_{i}$ and $σ_{i}\cap σ_{j}= \emptyset $ for all $i\ne j$. We say that $G$ is $σ$-primary if $G$ is a $σ_{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: $σ$-subnormal in $G$ if there is a subgroup chain $A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G$ such that either $A_{i-1}\trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $σ$-primary for all $i=1, \ldots, n$, modular in $G$ if the following conditions hold: …
On the subgroup separability of the free product of groups
本文研究了群的自由积的子群可分性问题，在剩余根类群的自由积框架下，证明了满足非平凡恒等式子群的根类可分性判定准则，并推广至群簇并集子群的可分性描述。
On the regularity of irreducible subgroups of finite classical groups
本文研究有限经典群中不可约子群的正则性，确定了非正则t元组的结构，证明当维度n≥15时，所有极大子群在S中都是正则的。
Overgroups of exterior powers of an elementary group. Normalizers
本文刻画了代数群概型 $\bigwedge^m GL_n$ 的几何与代数性质，并将其作为正规化子应用于初等子群上群的“三明治分类”及线性保持问题。
We give an easy proof of Schützenberger's Theorem stating that non-commutative formal power series are rational if and only if they are recognisable. A byproduct of this proof is a natural metric on a subgroup of invertible rational non-commutative power series. We describe a few features of this metric group.
If G and X are groups and N is a normal subgroup of X, then the G-closure of N in X is the normal subgroup X^G= bigcap{kerphi|phi:X-> G, with N subseteq kerphi} of X . In particular, 1^G = R_G X is the G-radical of X. Plotkin calls two groups G and H geometrically equivalent, written G H, if for any free group F of finite rank and any normal subgroup N of F the G-closure and the H-closure of N in F are the same. Quasiidentities are formulas of the form (bigwedge_{i<=n}w_i=1 -> w =1) for any words w, w_i (i<=n) in a free group. Generally geometrically equivalent groups satisfy the same quasiide…
For a finite group $G$ with a normal subgroup $H$, the normal subgroup based power graph of $G$, denoted by $Γ_H(G)$ whose vertex set $V(Γ_H(G))=(G\setminus H)\bigcup \{e\}$ and two vertices $a$ and $b$ are edge connected if $aH=b^mH$ or $bH=a^nH$ for some $m, n \in \mathbb{N}$. In this paper we obtain some fundamental characterizations of the normal subgroup based power graph. We show some relation between the graph $Γ_H(G)$ and the power graph $Γ(\frac{G}{H})$. We show that $Γ_H(G)$ is complete if and only of $\frac{G}{H}$ is cyclic group of order $1$ or $p^m$, where $p$ is prime number and …
In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group $G$ satisfying the Triple Product Property (TPP) as a means to bound the exponent $ω$ of matrix multiplication. Recently, Hedtke and Murthy discussed several methods to find TPP triples. Because the search space for subset triples is too large, it is only possible to focus on subgroup triples. We present methods to upgrade a given TPP triple to a bigger TPP triple. If no upgrade is possible we use reduction methods (based on random experiments and heuristic…
The notion of adequate subgroups was introduced by Jack Thorne. It is a weakening of the notion of big subgroup used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to prove some new lifting theorems. In an appendix to Thorne's article, it was shown in that certain groups were adequate. One of the key aspects was the question of whether the span of the semsimple elements in the group is the full endomorphism ring of an absolutely irreducible module. We show that this is the case in prime characteristic p for p-solv…
Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d,n be positive integers, and let N be a d-fold covering space of M. We show that the covering map induces an embedding of the n-th braid group B_n(M) of M in the (dn)-th braid group B_{dn}(N) of N, and give several applications of this result. First, we classify the finite subgroups of the n-th braid group of the real projective plane, from which we deduce an alternative proof of the classification of the finite subgroups of the mapping class group of the n-punctured real projective plan…
In this paper we partially answer a question of P. Tukia about the size of the difference between the big horospheric limit set and the horospheric limit set of a Kleinian group. We mainly investigate the case of normal subgroups of Kleinian groups of divergence type and show that this difference is of zero conformal measure by using another result obtained here: the Myrberg limit set of a non-elementary Kleinian group is contained in the horospheric limit set of any non-trivial normal subgroup.
We study two actions of big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. The first two parts of the paper are devoted to the definition of objects and tools needed to introduce these two actions; in particular, we define and prove the existence of equators for infinite type surfaces, we define the hyperbolic graph and the circle needed for the actions, and we describe the Gromov-boundary of the graph using the embedding of its vertices in the cir…
Open subgroups of $p$-adic algebraic groups
针对$p$-进代数群的开子群是否为半代数（即是否在$(\boldsymbol{\rm Q}_p, +, \times)$中可定义）的公开问题，证明了交换情形下答案为肯定，结合已有结果可推广至约化代数群。
We study algebraic closure and its relation with definable closure in free groups and more generally in torsion-free hyperbolic groups. Given a torsion-free hyperbolic group G and a nonabelian subgroup A of G, we describe G as a constructible group from the algebraic closure of A along cyclic subgroups. In particular, it follows that the algebraic closure of A is finitely generated, quasiconvex and hyperbolic. Suppose that G is free. Then the definable closure of A is a free factor of the algebraic closure of A and the rank of these groups is bounded by that of G. We prove that the algebraic c…
We generalise Merzlyakov's theorem about the first-order theory of non-abelian free groups to all acylindrically hyperbolic groups. As a corollary, we deduce that if $G$ is an acylindrically hyperbolic group and $E(G)$ denotes the unique maximal finite normal subgroup of $G$, then $G$ and the HNN extension $G\dot{\ast}_{E(G)}$, which is simply the free product $G\ast\mathbb{Z}$ when $E(G)$ is trivial, have the same $\forall\exists$-theory. As a consequence, we prove the following conjecture, formulated by Casals-Ruiz, Garreta and de la Nuez González: acylindrically hyperbolic groups have trivi…
The notion of adequate subgroups was introduced by Jack Thorne [42]. It is a weakening of the notion of big subgroups used in generalizations of the Taylor-Wiles method for proving the automorphy of certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] that if the dimension is small compared to the characteristic then all absolutely irreducible representations are adequate. Here we extend the result by showing that, in almost all cases, absolutely irreducible kG-modules in characteristic p, whose irreducible G+-sum…
Let $R$ be any associative ring with $1$, $n\ge 3$, and let $A,B$ be two-sided ideals of $R$. In the present paper we show that the mixed commutator subgroup $[E(n,R,A),E(n,R,B)]$ is generated as a group by the elements of the two following forms: 1) $z_{ij}(ab,c)$ and $z_{ij}(ba,c)$, 2) $[t_{ij}(a),t_{ji}(b)]$, where $1\le i\neq j\le n$, $a\in A$, $b\in B$, $c\in R$. Moreover, for the second type of generators, it suffices to fix one pair of indices $(i,j)$. This result is both stronger and more general than the previous results by Roozbeh Hazrat and the authors. In particular, it implies tha…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。