Fixed Subgroups of Endomorphisms of Free Products
Mihalis Sykiotis
设$G=\ast_{i=1}^{n}G_{i}$和$ϕ$是$G$的对称自同态.若$ϕ$是单态或$G$是一个单生成的剩余有限群，则$ϕ$的固定子群$Fix(ϕ)=\{g\in G:ϕ(g)=g\}$的Kurosh秩至多为$n$。
Let $G=\ast_{i=1}^{n}G_{i}$ and let $ϕ$ be a symmetric endomorphism of $G$. If $ϕ$ is a monomorphism or if $G$ is a finitely generated residually finite group, then the fixed subgroup $Fix(ϕ)=\{g\in G:ϕ(g)=g\}$ of $ϕ$ has Kurosh rank at most $n$.
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The Scale and Tidy Subgroups for Endomorphisms of Totally Disconnected Locally Compact Groups
全不连通的局部紧群$G$的自同态$α$的标度是最小指数$[α(U) : α(U)\cap U]$，对于$U$是$G$的紧开子群。建立了最小值子群的结构特征。这个特征将$α$的子群整齐的概念从以前理解的$α$是自同构的情况扩展到$α$仅仅是自同态的情况。
We prove that for any finitely generated relatively hyperbolic group G and any symmetric endomorphism f of G with relatively quasiconvex image, Fixf is relatively quasiconvex subgroup of G.
The concepts of the scale and tidy subgroups for an automorphism of a totally disconnected locally compact group were defined in seminal work by George A. Willis in the 1990s, and recently generalized to the case of endomorphisms (G. A. Willis, Math. Ann. 361 (2015), 403--442). We show that central facts concerning the scale, tidy subgroups, quotients, and contraction groups of automorphisms extend to the case of endomorphisms. In particular, we obtain results concerning the domain of attraction around an invariant closed subgroup.
On endomorphisms of profinite groups
在假设G的每个指数只有1/2个开子群的条件下（假设自动成立，例如，如果G是n-生成的），我们得到了关于投射有限群G的连续自同态的一些一般限制.特别地，给定这样一个群G和一个连续自同态φ，我们得到G的一个半直接分解为一个“收缩”的正规子群和一个补，在这个补上φ诱导一个自同构;正规子群和补都是闭的。若G同构于它自身的一个真开子群，则G有一个无穷阿贝尔正规pro-p子群.
Explicit bounds for fixed subgroups of endomorphisms of free products
本文研究自由积自同态的不动子群，旨在给出其秩的显式界，推广了 Bestvina 和 Handel 关于自由群自同构不动子群的 Scott 猜想。
We consider the subgroup of points of finite orbit through the action of an endomorphism of a virtually free group, with particular emphasis on the subgroup of eventually fixed points, EvFix($φ$): points whose orbit contains a fixed point. We provide an algorithm to compute the subgroup of fixed points of an endomorphism of a finitely generated virtually free group and prove that finite orbits have cardinality bounded by a computable constant, which allows us to solve several algorithmic problems: deciding if $φ$ is a finite order element of End($G$), if $φ$ is aperiodic, if EvFix($φ$) is fini…
Fixed points of endomorphisms of graph groups
对给定的图群$G$，证明了对于$G$的每个自同态$φ$，不动点子群Fix $\,φ$是可生成的当且仅当$G$是自由阿尔贝群的自由积.同样的条件也适用于周期点的子群。如果$G$的依赖图是一个传递森林，则对于自同构也得到了类似的结果。
Classification of Fixed Subgroups of Endomorphisms in Free-abelian Times Surface Groups
本文研究自由阿贝尔乘曲面群等群类的自同态性质，刻画其自同态双射条件，证明相关群的Hopf性，完成自同态不动子群完整分类并给出有限不动子群的等价条件。
Let G be a group and f be an endomorphism of G. A subgroup H of G is called f-inert if the meet of Hf and H has finite index in the image Hf. The subgroups that are f-inert for all inner automorphisms of G are widely known and studied in the literature, under the name inert subgroups. The related notion of inertial endomorphism, namely an endomorphism f such that all subgroups of G are f-inert, was introduced in [30] and thoroughly studied in [31, 33]. The dual notion of fully inert subgroup, namely a subgroup that is f-inert for all endomorphisms of an abelian group A, was introduced in [42] …
On the intersection of fixed subgroups of $F_n\times F_m$
本文研究 $F_n\ imes F_m$ 中固定子群的交集问题，证明对单同态与自同态可判定，一般情形未知，特定类型等价于波斯特对应问题。
Abelian $p$--groups and their endomorphism rings
本研究利用基数不变量系统，对阿贝尔$p$--群的全不变子群格和其自同态环的理想格完成了分类。
Growth rate of an endomorphism of a group
Bowen在[B]中定义了一个自同态的增长率的一个生成群，并将它与熵的映射$f:M \mapsto M$上的一个紧凑的流形。本文研究了有限生成群的自同态增长率的纯群论性质。我们表明，它是有限的，并有界的最大长度的图像的生成器。给出了一个等价的公式，它将自同态的增长率与子群的增长链联系起来。然后，我们考虑了整群上自同态的增长率与仅限于子群或商上的增长率之间的关系，并利用这些结果计算了直积和半直积的增长率.然后，我们计算了几个不同类别的群，包括阿贝尔和幂零的自同态的增长率。
Lie groups over local fields furnish prime examples of totally disconnected, locally compact groups. We discuss the scale, tidy subgroups and further subgroups (like contraction subgroups) for analytic endomorphisms of such groups. The text is both a research article and a worked out set of lecture notes for a mini-course held June 27-July 1, 2016 at the MATRIX research center in Creswick (Australia) as part of the "Winter of Disconnectedness". The text can be read in parallel to the earlier lecture notes arXiv:0804.2234 which are devoted to automorphisms, with sketches of proof. Complementary…
We give an algorithm which computes the fixed subgroup and the stable image for any endomorphism of the free group of rank two $F_2$, answering for $F_2$ a question posed by Stallings in 1984 and a question of Ventura.
Endomorphism rings of permutation modules over maximal Young subgroups
令$K$是特征为2的域，令$λ$是某个自然数$r$的两部分划分。将对应于$Σ_r$中的（最大）Young子群$Σ_λ$的置换模表示为$M^λ$。构造了Schur代数$S_K(2,r)$的中心化子代数$S_K(λ) = 1_λS_K(2,r) 1_λ= End_{KΣ_r}(M^λ)$的正交本原幂等元的一个全集.这些幂等元自然地与2-Kostka数一一对应。
The group fixed by a family of endomorphisms of a surface group
对于具有$χ(S)<0$的闭曲面$S$，证明了$π_1(S)$的自同态族$\mathcal B$的不动点子群具有$\rk \fix\mathcal B\leq \rk π_1(S)$ .特别地，如果$\mathcal B$包含非满态自同态，则$\rk \fix\mathcal B\leq \frac{1}{2} \rk π_1(S)$。我们还证明了$π_1(S)$的几何子群是惰性的，因此$π_1(S)$的满射族的固定子群也是惰性的。
We study the scale and tidy subgroups of an endomorphism of a totally disconnected locally compact group using a geometric framework. This leads to new interpretations of tidy subgroups and the scale function. Foremost, we obtain a geometric tidying procedure which applies to endomorphisms as well as a geometric proof of the fact that tidiness is equivalent to being minimizing for a given endomorphism. Our framework also yields an endomorphism version of the Baumgartner-Willis tree representation theorem. We conclude with a construction of new endomorphisms of totally disconnected locally comp…
On the ring of inertial endomorphisms of an abelian group
称阿贝尔群$A$的自同态$φ$为惯性的，如果$A$的每个子群$H$在$H+φ(H)$中有有限的指数.研究了阿贝尔群的惯性自同态环。在这里，我们得到了一个令人满意的描述模的理想有限自同态。也被认为是向量空间的相应问题。关于阿贝尔群的惯性自同态的特征，见arXiv：1310.4625。可逆惯性自同态群在arXiv：1403.4193中已经研究过。
We prove that ascending HNN extensions of free groups are word-hyperbolic if and only if they have no Baumslag-Solitar subgroups. This extends the theorem of Brinkmann that free-by-cyclic groups are word-hyperbolic if and only if they have no free abelian subgroups of rank 2. The paper is split into two independent parts: 1) We study the dynamics of injective nonsurjective endomorphisms of free groups. We prove a canonical structure theorem that initializes the development of improved relative train tracks for endomorphisms; this structure theorem is of independent interest since it makes many…
The interplay between set-theoretic solutions of the Yang--Baxter equation of Mathematical Physics, skew braces, regular subgroups, and Hopf--Galois structures has spawned a considerable body of literature in recent years. In a recent paper, Alan Koch generalised a construction of Lindsay N.~Childs, showing how one can obtain bi-skew braces $(G, \cdot, \circ)$ from an endomorphism of a group $(G, \cdot)$ whose image is abelian. In this paper, we characterise the endomorphisms of a group $(G, \cdot)$ for which Koch's construction, and a variation on it, yield (bi-)skew braces. We show how the s…
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 describe the endomorphisms of the direct product of two free groups of finite rank and obtain conditions for which the subgroup of fixed points is finitely generated and we do the same for periodic points. We also describe the automorphisms of such a group and solve the three Whitehead problems for automorphisms, monomorphisms and endomorphisms for groups in this class. We also obtain conditions for an endomorphism to be uniformly continuous when a suitable metric is considered and study infinite fixed points of the extension to the completion of the group.
Large mapping tori of free group endomorphisms
我们提出了一个算法，任何有限的介绍一组作为输入，将终止与答案是当且仅当该组是大的。利用这一点，我们证明了一个映射环面的一个新的生成的自由群自同构是大的，如果它包含整数倍的整数作为一个子群的无限指数。然后，我们将这一结果推广到映射环面的n-生成的自由群自同态，以及表明，这样的一个组是大的，如果它包含一个Baumslag-Solitar群的无限指数，并有一个有限的指数子群的第一贝蒂数至少为2。我们还表明，如果一个群体拥有一个不足1介绍其中一个relator是一个换向器，那么它是整数倍的整数，或者它是大的，或者它是尽可能从剩余有限。
Computation of endo-fixed closures in free-abelian times free groups
本文研究自由阿贝尔乘自由群自同态不动子群的行为，提出一种结合组合与代数方法的算法，用于判定有限生成子群是否为某自同态族的不动子群并构造该族。
We give an explicit characterization of which direct products $G$ of surface groups of Euclidean type satisfy that the fixed subgroup of any automorphism (or endomorphism) of $G$ is compressed, and of which is it always inert.
Mapping tori of free group automorphisms are coherent
群G的一个自同态Φ的映射环面是HNN-扩张G*_G，其键映射是单位映射和Φ。证明了内射自由群自同态的映射环面具有其生成子群是双表示的性质，并且这些子群是有限型的。
Homomorphisms of commutator subgroups of braid groups with small number of strings
本文刻画任意股数辫子群 $B_n$ 及其换位子群 $B'_n$ 的自同态与相关同态，补足小股数情形，并对 $n=4$ 与 $n\ge5$ 使用不同方法。
Endomorphisms of relatively hyperbolic groups
本文将Paulin和Rips-Sela关于双曲群自同态的一些结果推广到相对双曲群，特别证明了以下结果。(1)若G是一个具有细长抛物子群的非初等相对双曲群，且G不是余霍普夫群或Out（G）是无限的，则G在一个细长群上分裂. (2)若非初等相对双曲群的非抛物生成的非抛物子群H不是霍普夫子群，则H非平凡地作用在R-树上. (3)每一个非初等相对双曲群都有一个非初等相对双曲商是霍普夫的。(4)对于某个Kazhdan群H，任何一个非线性群同构于Out（H）的一个有限指数子群. (This[001 pdf 1st-31 files]锐化奥利维尔-怀斯的结果）。
On the irreducibility of Deligne-Lusztig varieties
设$G$是定义在有限域的代数闭包上的连通约化代数群，$F : G \to G$是一个自同态，使得$F^d$是某个$d \geq 1$的Frobenius自同态.设$P$是$G$的一个抛物子群，允许一个$F$ -稳定的Levi子群.证明了Deligne-Lusztig簇$\{gP | g^{-1}F(g)\in P\cdot F(P)\}$是不可约的当且仅当$P$不包含在$G$的真$F$ -稳定抛物子群中.
We prove that the property of a free group endomorphism being irreducible is a group invariant of the ascending HNN extension it defines. This answers a question posed by Dowdall-Kapovich-Leininger. We further prove that being irreducible and atoroidal is a commensurability invariant. The invariance follows from an algebraic characterization of ascending HNN extensions that determines exactly when their defining endomorphisms are irreducible and atoroidal; specifically, we show that the endomorphism is irreducible and atoroidal if and only if the ascending HNN extension has no infinite index s…
On fixed points and stabilizers in solvable Baumslag--Solitar groups
本文研究了可解Baumslag-Solitar群中的固定点和稳定化子，证明了自同构和满射下的固定点子群及稳定化子的结构特性。
The Pinsker subgroup of an algebraic flow
定义在阿贝尔群G的自同态f上的代数熵h度量G的非空有限子集F关于f的轨道的增长。我们表明，这种增长可以是多项式或指数。G的最大f-正规子群与G的最大f-正规子群P（G，f）重合，使得h（f|_P（G，f））=0。从f的拟周期点的观点出发，我们还得到了P（G，f）的另一个特征.这给出了遍历论中的如下应用：对于紧阿贝尔群K的每个连续内射自同态g，存在K的最大g-不变闭子群N，使得g|_N是遍历的，并且商K/N的诱导自同态g'的拓扑熵为零.
We prove that the mapping torus of a graph immersion has a word-hyperbolic fundamental group if and only if the corresponding endomorphism does not produce Baumslag-Solitar subgroups. Due to a result by Reynolds, this theorem applies to all injective endomorphisms of $F_2$ and nonsurjective fully irreducible endomorphisms of $F_n$. We also give a framework for extending the theorem to all injective endomorphisms of $F_n$.
Automorphism tower problem and semigroup of endomorphisms for free Burnside groups
证明了自由Burnside群$B(m,n)$的所有内自同构的群是$Aut(B(m,n))$的所有子群中唯一的正规子群，对于所有奇数$n\ge1003$和$m>1$，这些子群同构于具有某秩$s$的自由Burnside群$B(s,n)$.自由Burnside群$B(m,n)$的自同构群$Aut(B(m,n))$对于奇$n\ge1003$是完备的，即它有平凡中心，并且$Aut(B(m,n))$的任何自同构都是内自同构。因此，对于群$B(m,n)$解决了自同构塔问题，并证明了它与绝对自由群的自同构塔一样短。证明了$End(B(m,n))$的每个自同构都是由$Aut(B(m,n))$的元素共轭的.
Virtual Endomorphisms of Nilpotent Groups
群G的虚自同态是从H到G的同态f，其中H是G的有限指数m的子群。三元组（G，H，f）在单根m元树上产生G的状态封闭（或自相似）表示t。本文研究了当G是幂零的时象G^t的性质。特别地，证明了若G是无挠生成的幂零群，则G^t的可解度以m的素除子个数为界.
On Eigenvalues of Free Group Endomorphisms
设F是一个F生成的自由群F的自同态，H是F的一个有限指标子群，在F下是不变的。H的非零特征值包含在H限制的特征值中。
On the intersection of subgroups in free groups: echelon subgroups are inert
自由群$F$的子群$H$称为在$F$中惰性，如果对于每个$G < F$，$H$与$G$的交集的秩不大于$G$的秩。本文推广了已知的惰性子群族。证明了1-生成元自同态的惯性性质。自由群中的等差、梯子群是惰性的。梯队子群是通过一组生成元定义的，这些生成元相对于自由群的某个有序基呈梯队形式，并且可以被视为自由因子的推广。例如，n-生成的自由群的自同构的固定子群是梯队子群。证明主要遵循图论或组合方法。
A finitely generated group $Γ$ is called strongly scale-invariant if there exists an injective endomorphism $φ: Γ\to Γ$ with the image $φ(Γ)$ of finite index in $Γ$ and the subgroup $\displaystyle \bigcap_{n>0}φ^n(Γ)$ finite. The only known examples of such groups are virtually nilpotent, or equivalently, all examples have polynomial growth. A question by Nekrashevych and Pete asks whether these groups are the only possibilities for such endomorphisms, motivated by the positive answer due to Gromov in the special case of expanding group morphisms. In this paper, we study this question for the …
Free idempotent generated semigroups and endomorphism monoids of free $G$-acts
对双序集$E$上的自由幂等生成半群$\mathrm{IG}(E)$的研究始于20世纪70年代Nambooripad的开创性工作，最近又出现了一些新的方法，包括几何方法和组合方法。本文研究了$\mathrm{IG}(E)$在$E$是环积$G\wr \mathcal{T}_n$的双序集的情况下的性质，其中$G$是群，$\mathcal{T}_n$是$n$元素上的全变换幺半群.这个环积同构于$n$生成元上的自由$G$ -act $F_n(G)$的自同态幺半群，这为我们提供了一个方便的途径。我们说$F_n(G)$的元素的秩是它的图像中（自由）生成器的最小数量。令$\varepsilon=\varepsilon^2\in F_n(G).$由于相当直接的原因，已知如果$\mathrm{rank}\,\varepsilon =n-1$（分别为$n$），则包含$\varepsilon$的$\mathrm{IG}(E)$的极大子群是自由的（分别为平凡的）。我们证明了如果$\mathrm{rank}\,\varepsilon =r$，其中$1\leq r\leq n-2$，那么包含$\varepsilon$的$\mathrm{IG}(E)$的极大子群同构于$F_n(G)$中的极大子群，从而同构于$G\wr \mathcal{S}_r$，其中$\mathcal{S}_r$是$r$元素上的对称群。我们之前…
Computing fixed closures in free groups
设$F$是一个随机生成的自由群。我们提出了一个算法，这样，给定一个子群$H\leqslant F$，决定是否$H$是固定的子群的一些家庭的自同构，或家庭的自同态$F$，并在肯定的情况下，发现这样的家庭。该算法结合了组合方法和几何方法。
Determinant for automorphisms of semidirect product of groups
本文利用半直积自同态的 $2\ imes2$ 映射矩阵表示，引入并研究其行列式，进而用行列式工具刻画可逆自同态。
Hölder conditions for endomorphisms of hyperbolic groups
证明了双曲群$G$的自同态$φ$满足关于可视度量的赫尔德条件当且仅当$φ$是虚内射的且$Gφ$是$G$的拟凸子群.如果$G$是虚自由的或无挠的余霍普夫的，则$φ$是一致连续的当且仅当它满足赫尔德条件当且仅当它是虚内射的。讨论了自由群自同构的利普希茨条件。
We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on such functional equations for ideal zeta functions of nilpotent Lie lattices. Via the Mal'cev correspondence, these results have corollaries pertaining to zeta functions enumerating normal subgroups of finite index in finitely generated nilpotent groups, most notably finitely generated free nilpotent groups of any given class.
Fixed subgroups are compressed in surface groups
对于紧致曲面$Σ$（可定向或不可定向，有边界或无边界），我们证明了$π_1(Σ)$的自同态的任何族$B$的固定子群$\operatorname{Fix} B$在$π_1(Σ)$中被压缩，即，任何子组$\operatorname{Fix} B \leq H \leq π_1(Σ)$的$\operatorname{rk}((\operatorname{Fix} B)\cap H)\leq \operatorname{rk}(H)$。在此基础上，我们给出了惯性猜想的部分正解，包括自由群和曲面群。我们还研究了许多自由基和表面基团的直积$G$，并给出了当$G$对每一个$ϕ\in Aut(G)$都满足$\operatorname{rk}(\operatorname{Fix} ϕ) \leq \operatorname{rk}(G)$时的一个特征.
Fixed points of endomorphisms of virtually free groups
证明了逆变换器的一个不动点定理，从而给出了一个逆生成的虚自由群的自同态的不动点子群是逆生成的自动机理论证明.若双曲度量的自同态是一致连续的，则证明了双曲边界上的正则不动点集在有限个不动点的作用下有2个轨道.在自同构情形下，证明了这些正则不动点是指数稳定的吸引子或指数稳定的排斥子。
Finiteness results for subgroups of finite extensions
本文在有限扩张的背景下讨论了Takahasi和Howson关于自由群子群的两个经典定理。我们提供了范围内的子群的交集类的群体，如虚拟自由群，虚拟幂零群或基本群的有限图形的群体与虚拟多环顶点群和有限棱道群的秩。作为Takahasi定理推广的一个应用，我们给出了有限群图的基本群的任意自同态的周期子群的秩的一个一致界，该图的群具有n-生成的虚幂零顶点群和有限棱道群.
On the dynamics of endomorphisms of the direct product of two free groups
本文证明了$F_n\times F_m$自同态的Brinkmann问题及单射自同态的双边共轭问题可判定，并证明了其自同构在无穷远处的动力学具有渐近周期性。
The classical result by Dyer--Scott about fixed subgroups of finite order automorphisms of $F_n$ being free factors of $F_n$ is no longer true in $Z^m\times F_n$. Within this more general context, we prove a relaxed version in the spirit of Bestvina--Handel Theorem: the rank of fixed subgroups of finite order automorphisms is uniformly bounded in terms of $m,n$. We also study periodic points of endomorphisms of $Z^m\times F_n$, and give an algorithm to compute auto-fixed closures of finitely generated subgroups of $Z^m\times F_n$. On the way, we prove the analog of Day's Theorem for real eleme…
Determinant for automorphisms of Zappa-Szép product of groups
本文利用映射的2×2矩阵描述，研究两个群Zappa-Szép积的自同态行列式，刻画可逆自同态，并计算交错群A5的行列式。
We obtain conditions of uniform continuity for endomorphisms of free-abelian times free groups for the product metric defined by taking the prefix metric in each component and establish an equivalence between uniform continuity for this metric and the preservation of a coarse-median, which was recently introduced by Fioravanti. Considering the extension of an endomorphism to the completion we count the number of orbits for the action of the subgroup of fixed points (resp. periodic) points on the set of infinite fixed (resp. periodic) points. Finally, we study the dynamics of infinite points: f…
A note on subgroups of automorphism groups of full shifts
讨论了满移位的自同构群的子群集及其自同态幺半群的子幺半群。证明了么半群在直积下的闭包和群在自由积下的闭包。我们还证明了全移位的自同构群嵌入不可数sofic移位的自同构群。作为推论，得到了一些不可判定性结果。
Let $G$ be a countable monoid and let $A$ be an Artinian group (resp. an Artinian module). Let $Σ\subset A^G$ be a closed subshift which is also a subgroup (resp. a submodule) of $A^G$. Suppose that $Γ$ is a finitely generated monoid consisting of pairwise commuting cellular automata $Σ\to Σ$ that are also homomorphisms of groups (resp. homomorphisms of modules) with monoid binary operation given by composition of maps. We show that the valuation action of $Γ$ on $Σ$ satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissib…
Relative hyperbolicity of ascending HNN extension of groups
该研究证明了带自由因子系统的有限生成群的保系统单自同态对应的升序HNN扩张的相对双曲性，并得出有限秩自由群指数增长单自同态的升序HNN扩张相对双曲的推论，推进了几何群论相关研究。
Iwip endomorphisms of free groups and fixed points of graph selfmaps
本文通过研究自由群的iwip外自同态在稳定树上的作用，给出了Jiang-Wang-Zhang关于图自映射不动点类量的下界达到等号的充分条件，部分回答了Jiang的问题。
Surface subgroups from linear programming
我们发现，某些类的自由群的图形包含表面子群，包括团体与积极的$b_2$通过加倍自由群沿收集的子群，并通过“随机”上升的HNN扩展的自由群获得的团体。一个特例是与秩为2的自由群的自同态相关联的HNN扩张，它将a发送到ab，将b发送到ba;这个例子（以及随机例子）回答了萨皮尔的负问题。我们进一步证明了单位球的Gromov范数（在2维）的一个双重的自由群沿收集的子群是一个有限边的合理多面体，和每个合理的类是虚拟的极值曲面子群表示。这些结果是通过组合、几何和线性规划技术的混合获得的。
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