Serge Cantat, Vincent Guirardel, Anne Lonjou
Consider an algebraically closed field k and the Cremona group of all birational transformations of the projective plane over k. We characterize infinite order elements of this group having a non-zero power generating a proper normal subgroup of the Cremona group.
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We classify all finite simple subgroups in the Cremona group of rank 3
The Cremona group is the group of birational transformations of the complex projective plane. In this paper we classify its subgroups that consist only of elliptic elements using elementary model theory. This yields in particular a description of the structure of torsion subgroups. As an appliction, we prove the Tits alternative for arbitrary subgroups of the Cremona group, generalizing a result of Cantat. We also describe solvable subgroups of the Cremona group and their derived length, refining results from Déserti.
We give some examples of non-nilpotent locally nilpotent, and hence nonlinear subgroups of the planar Cremona group.
Linearization problem for finite subgroups of the plane Cremona group
该论文解决了特征零代数闭域上平面Cremona群有限子群的线性化问题，提供了完整解法。
We give the classification of elements - respectively cyclic subgroups - of finite order of the Cremona group, up to conjugation. Natural parametrisations of conjugacy classes, related to fixed curves of positive genus, are provided.
We complete the classical and modern work on the classification of conjugacy classes of finite subgroups of the group of birational transformations of the complex projective plane.
In this paper we describe conjugacy classes of finite subgroups of odd order in the group of birational automorphisms of the real projective plane.
Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane P^2 over k is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory, and algebraic geometry to produce elements in the Cremona group that generate non trivial normal subgroups.
Finitely generated subgroups of algebraic elements of plane Cremona groups are bounded
该研究证明平面Cremona群中仅含代数元素的有限生成子群次数有界，基于无穷直和上“良好”作用的更一般结论，还刻画了这类子群的次数增长特性。
Exploring Bass' Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass' Triangulability Problem is the affirmative. To this end we prove a theorem on invariant subfields of $1$-extensions. We also obtain a general construction of all rationally triangulable subgroups of the Cremona groups and, as an application, classify rationall…
Representations of finite subgroups of Cremona groups
本文研究 Cremona 群有限子群忠实表示的最小维数，给出秩 1、2 的精确值，并证明正特征高秩情形为无穷大、特征 0 情形有限且下界随秩指数增长。
This work presents the conjugacy classes of finite abelian subgroups of the Cremona group of the plane. Using a well-known theory, this problem amounts to the study of automorphism groups of some Del Pezzo surfaces and conic bundles. We have thus to enumerate all the cases, which gives a long description, and then to show whether two cases are distinct or not, using some conjugacy invariants. For example, we use the non-rational curves fixed by one element of the group, and the action of the whole group on these curves. From this classification, we deduce a sequence of more general results on …
We study automorphism groups of real del Pezzo surfaces, concentrating on finite groups acting minimally on them. As a result, we obtain a vast part of classification of finite subgroups in the real plane Cremona group.
In this paper, we show that Cremona groups are sofic. We actually introduce a quantitative notion of soficity, called sofic profile, and show that the group of birational transformations of a d-dimensional variety has sofic profile at most polynomial of degree d. We also observe that finitely generated subgroups of the Cremona group have a solvable word problem. This provides examples of finitely generated groups with no embeddings into any Cremona group, answering a question of S. Cantat.
The essential and Cremona dimensions of a group
本文通过实例支持猜想：有限群在复数域上的克雷莫纳维数不大于其本质维数。
We explore algebraic subgroups of of the Cremona group $\mathcal C_n$ over an algebraically closed field of characteristic zero. First, we consider some class of algebraic subgroups of $\mathcal C_n$ that we call flattenable. It contains all tori. Linearizability of the natural rational actions of flattenable subgroups on the affine space $\An$ is intimately related to rationality of the invariant fields and, for tori, is equivalent to it. We prove stable linearizability of these actions and show the existence of nonlinearizable actions among them. This is applied to exploring maximal tori in …
We describe the nilpotent subgroups of the group Bir(P^2(C)) of birational transformations of the complex projective plane. Let N be a nilpotent subgroup of class k>1; then either each element of N has finite order, or N is virtually metabelian.
The first group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in other groups. The third concerns the connectedness of the Cremona groups.
We study the algebraic structure of the $n$-dimensional Cremona group and show that it is not an algebraic group of infinite dimension (ind-group) if $n\ge 2$. We describe the obstruction to this, which is of a topological nature. By contrast, we show the existence of a Euclidean topology on the Cremona group which extends that of its classical subgroups and makes it a topological group.
Consider a Cremona group endowed with the Euclidean topology introduced by Blanc and Furter. It makes it a Hausdorff topological group that is not locally compact nor metrisable. We show that any sequence of elements of the Cremona group of bounded order that converges to the identity is constant. We use this result to show that the Cremona groups do not contain any non-trivial sequence of subgroups converging to the identity. We also show that, in general, paths in a Cremona group do not lift and do not satisfy a property similar to the definition of morphisms to a Cremona group.
We show that plane Cremona groups over finite fields embed as dense subgroups into Neretin groups, i.e. groups of almost automorphisms of rooted trees. We also show that if the finite base field has even characteristic and contains at least 4 elements, then the permutations induced by birational transformations on rational points of regular projective surfaces are even. In a second part, we construct explicit locally compact CAT(0) cube complexes, on which Neretin groups act properly. This allows us to recover in a unified way various results on Neretin groups such as that they are of type $F_…
Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least 3 factors through a homomorphism onto a cocompact discrete subgroup of PSL(2,R). We also study actions of Kähler groups on infinite dimensional real hyperbolic spaces, describe some exotic actions of PSL(2,R) on these spaces, and give an application to the study of the Cremona group.
We use a recent advance in birational geometry to prove new lower bounds on the essential dimension of some finite groups.
We prove that the group of birational transformations of a Del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2. As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, its subgroup generated by all connected algebraic subgroups is a proper normal subgroup.
Birational maps of Severi-Brauer surfaces, with applications to Cremona groups of higher rank
本研究通过刻画非平凡Severi-Brauer曲面的双有理变换群性质，推导得出秩≥4的Cremona群的商群结构、生成元性质、Hopfian性质及3-挠元不可数等关键群论结论。
Using a theorem of Dahmani, Guirardel and Osin we prove that the Cremona group in 2 dimension is not simple, over any field. More precisely, we show that some elements of this group satisfy a weakened WPD property which is equivalent in our particular context to the Bestvina and Fujiwara's one.
Geometric and dynamic properties of embeddings of SL(2,Z) into the Cremona group are studied. Infinitely many non-conjugate embeddings which preserve the type (i.e. which send elliptic, parabolic and hyperbolic elements onto elements of the same type) are provided. The existence of infinitely many non-conjugate elliptic, parabolic and hyperbolic embeddings is also shown. In particular, a group G of automorphisms of a smooth surface S obtained by blowing-up 10 points of the complex projective plane is given. The group G is isomorphic to SL(2,Z), preserves an elliptic curve and all its elements …
On cube and Cremona rigidity for higher-rank lattices
研究实秩至少为2的半单李群中不可约格在CAT(0)立方体复形上的作用，通过证明上同调消失结果及利用超刚性等方法，得出相关作用可正则性的结论，绕过对FW性质的直接使用。
We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups, $Out(F_n)$, and the Cremona group. Other examples can be found among groups acting geometrically on $CAT(0)$…
We introduce the notion of über-contracting element, a strengthening of the notion of strongly contracting element which yields a particularly tractable criterion to show the acylindrical hyperbolicity, and thus a strong form of non-simplicity, of groups acting on non locally compact spaces of arbitrary dimension. We also give a simple local criterion to construct über-contracting elements for groups acting on complexes with unbounded links of vertices. As an application, we show the acylindrical hyperbolicity of the tame automorphism group of $\mathrm{SL}_2(\mathbb{C})$, a subgroup of the $3$…
We give a description of a finite group whose maximal subgroups possess only soluble proper subgroups, which implies the answer to the well-known question on composition factors of finite groups, whose second maximal subgroups are soluble.
Finite groups in which every cyclic subgroup is the intersection of maximal subgroups
本文刻画了每个循环子群均为极大子群交集的有限群的结构，并与所有真子群均为极大子群交集的性质进行对比。
In the paper we prove that the Hall property $C_π$ is inherited by normal subgroups which index is a $π'$-number.
Following J.S. Rose, a subgroup H of a group G is said contranormal in G if G = H^G . In a certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. We prove that a nilpotent-by-finite group with no proper contranormal subgroup is nilpotent. There are locally nilpotent groups with a proper contranormal subgroup. We study the structure of hypercentral groups with a finite proper contranormal subgroup.
In this paper we provide a sufficient condition for a subgroup of a free group not to have the almost congruence extension property (ACEP). We also show that any finitely generated subgroup of a free group satisfies some a generalization of ACEP.
Let $k$ be a positive integer and $G$ be a finite group that cannot be written as the union of $k$ proper subgroups. In this short note, we study the existence of a constant $c_k\in (0,1)$ such that $|\cup_{i=1}^k H_i| \leq c_k|G|$, for all proper subgroups $H_1$, ..., $H_k$ of $G$.
We determine all the ways in which a direct product of two finite groups can be expressed as the set-theoretical union of proper subgroups in a family of minimal cardinality.
Given an action of a group G on a topological space X, we establish a necessary and sufficient condition for the existence of a free subgroup F of rank 2 of G acting properly discontinuously on at least one nonempty, open, F-invariant subspace of X. In the case of a discrete topology (group action on a set X), the condition simply detects free subgroups of rank 2 acting freely on some orbit.
In the context of group-theoretic fast matrix multiplication the TPP capacity is used to bound the exponent $ω$ of matrix multiplication. We prove a new and sharper upper bound for the TPP subgroup capacity of a finite group
We show how to count and randomly generate finitely generated subgroups of the modular group $\textsf{PSL}(2,\mathbb{Z})$ of a given isomorphism type. We also prove that almost malnormality and non-parabolicity are negligible properties for these subgroups. The combinatorial methods developed to achieve these results bring to light a natural map, which associates with any finitely generated subgroup of $\textsf{PSL}(2,\mathbb{Z})$ a graph which we call its silhouette, and which can be interpreted as a conjugacy class of free finite index subgroups of $\textsf{PSL}(2,\mathbb{Z})$.
Subgroup induction property for branch groups
本文研究了分支群的子群归纳性质，证明了具有该性质的有限生成分支群必为挠群、刚好无限且子群可分，并证明所有挠GGS群均具有此性质，提供了首个无限族例子。
Given a finite non-cyclic group $G$, call $σ(G)$ the least number of proper subgroups of $G$ needed to cover $G$. In this paper we give lower and upper bounds for $σ(G)$ for $G$ a group with a unique minimal normal subgroup $N$ isomorphic to $A_n^m$ where $n \geq 5$ and $G/N$ is cyclic. We also show that $σ(A_5 \wr C_2)=57$.
A valuated group with normal forms is a group with an integer-valued length function satisfying some Lyndon's axioms and an additional axiom considered by Hurley. We prove a subgroup theorem for valuated groups with normal forms analogous to Grushko-Neumann's theorem. We study also the CSA property in such groups.
A Schmidt group is a non-nilpotent group whose every proper subgroup is nilpotent. We study the properties of a non-nilpotent group G in which every Schmidt subgroup is a Hall subgroup of G.
Given a finite non-cyclic group $G$, call $σ(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $σ(G) < σ(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is \textit{monolithic}, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.
We describe some of the geometric properties of the automorphism group Aut(F) of Thompson's group F. We give realizations of Aut(F) geometrically via periodic tree pair diagrams, which lead to natural presentations and give effective methods for estimating the word length of elements. We study some natural subgroups of Aut(F) and their metric properties. In particular, we show that the subgroup of inner automorphisms of F is at least quadratically distorted in Aut(F), whereas other subgroups of Aut(F) isomorphic to F are undistorted.
A residually finite (profinite) group $G$ is just infinite if every non-trivial (closed) normal subgroup of $G$ is of finite index. This paper considers the problem of determining whether a (closed) subgroup $H$ of a just infinite group is itself just infinite. If $G$ is not virtually abelian, we give a description of the just infinite property for normal subgroups in terms of maximal subgroups. In particular, we show that such a group $G$ is hereditarily just infinite if and only if all maximal subgroups of finite index are just infinite. We also obtain results for certain families of virtual…
For a non-cyclic finite group $X$ let $σ(X)$ be the least number of proper subgroups of $X$ whose union is $X$. Precise formulas or estimates are given for $σ(S \wr C_{m})$ for certain nonabelian finite simple groups $S$ where $C_m$ is a cyclic group of order $m$.
The Sylow p-subgroups of the symmetric group S_p^n satisfy the appropriate generalization of Maschke's Theorem to the case of a p'-group acting on a (not necessarily abelian) p-group. Moreover, some known results about the Sylow p-subgroups of S_p^n are stated in a form that is true for all primes p.
We show that the class of amalgamated free products of two free groups over a cyclic subgroup admits amenable, faithful and transitive actions on infinite countable sets. This work generalizes the results on such actions for doubles of free group on 2 generators.
Let $π$ be a set of primes. We say that a finite group $G$ is a $D_π$-group if all its maximal $π$-subgroups are conjugate. In this paper, we give an affirmative answer to the problem 17.44(b) from "The Kourovka notebook": we prove that in a $D_π$-group an overgroup of a $π$-Hall subgroup is always a $D_π$-group.
Multi-GGS-groups have the congruence subgroup property
本文将 GGS-groups 的同余子群性质推广至 multi-GGS-groups 族，指出除常向量定义者外均具此性质，并强调推广证明需新思路。
Let $π$ be a set of primes. According to H. Wielandt, a subgroup $H$ of a finite group $X$ is called a $π$-submaximal subgroup if there is a monomorphism $ϕ:X\rightarrow Y$ into a finite group $Y$ such that $X^ϕ$ is subnormal in $Y$ and $H^ϕ=K\cap X^ϕ$ for a $π$-maximal subgroup $K$ of $Y$. In his talk at the well-known conference on finite groups in Santa-Cruz (USA) in 1979, Wielandt posed a series of open questions and among them the following problem: to describe the $π$-submaximal subgroup of the minimal nonsolvable groups and to study properties of such subgroups: the pronormality, the in…
Suppose that $G$ is a group, $H$ and $K$ are proper isomorphic central subgroups of $G$, and $\mathfrak{G}$ is an HNN-extension of $G$ with the associated subgroups $H$ and $K$. We prove necessary and sufficient conditions for $\mathfrak{G}$ to be residually a $\mathcal{C}$-group, where $\mathcal{C}$ is a class of groups closed under taking subgroups, extensions, homomorphic images, and Cartesian products of the form $\prod_{y \in Y}X_{y}$, where $X, Y \in \mathcal{C}$ and $X_{y}$ is an isomorphic copy of $X$ for each $y \in Y$.
We compute the relative divergence and the subgroup distortion of Bestvina-Brady subgroups. We also show that for each integer $n\geq 3$, there is a free subgroup of rank $n$ of some right-angled Artin group whose inclusion is not a quasi-isometric embedding. This result answers the question of Carr about the minimum rank $n$ such that some right-angled Artin group has a free subgroup of rank $n$ whose inclusion is not a quasi-isometric embedding. It is well-known that a right-angled Artin group $A_Γ$ is the fundamental group of a graph manifold whenever the defining graph $Γ$ is a tree. We sh…
Finite groups whose proper subgroups of order divisible by $p$ are supersolvable
研究阶可被奇素数p整除且真子群为超可解但自身非超可解的有限群，通过探讨可解与不可解情形，来分析这类群的算术及结构性质。
For a finite group $G$ let $σ(G)$ (the "sum" of $G$) be the least number of proper subgroups of $G$ whose set-theoretical union is equal to $G$, and $σ(G)=\infty$ if $G$ is cyclic. We say that a group $G$ is $σ$-elementary if for every non-trivial normal subgroup $N$ of $G$ we have $σ(G)<σ(G/N)$. In this paper we produce the list of all the $σ$-elementary groups of sum up to 25. We also show that $σ(\Aut(PSL(2,8)))=29$.
A group G has property FA if G fixes a point of every tree on which G acts without inversions. We prove that every Coxeter system of finite rank has a visual JSJ decomposition over subgroups with property FA. As an application, we reduce the twist conjecture to Coxeter systems that are indecomposable with respect to amalgamated products over visual subgroups with property FA.
Let $G$ be a group. We define the coprime graph of subgroups of $G$, denoted by $\mathcal P(G)$, is a graph whose vertex set is the set of all proper subgroups of $G$, and two distinct vertices are adjacent if and only if the order of the corresponding subgroups are coprime. In this paper, we study some connections between algebraic properties of a group and graph theoretic properties of its coprime graph.
A $p$-subgroup $H$ of a finite group $G$ is said to satisfy partial $S$-$Π$-property in $G$ if $G$ has a chief series $Γ_{G}: 1=G_{0}<G_{1}<\cdots<G_{n}=G$ such that for every $G$-chief factor $G_{i}/G_{i-1}$ $(1\leqslant i\leqslant n)$ of $Γ_{G}$, either $(H\cap G_{i})G_{i-1}/G_{i-1}$ is a Sylow $p$-subgroup of $G_{i}/G_{i-1}$ or $|G/G_{i-1}: N_{G/G_{i-1}}((H\cap G_{i})G_{i-1}/G_{i-1})|$ is a $p$-number. In this paper, we mainly investigate the structure of finite groups with some primary subgroups satisfying partial $S$-$Π$-property.
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