We show that a finitely generated residually finite rationally solvable (or RFRS) group $G$ is virtually fibred, in the sense that it admits a virtual surjection to $mathbb{Z}$ with a finitely generated kernel, if and only if the first $L^2$-Betti number of $G$ vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of $3$-manifolds.
{"title":"Residually finite rationally solvable groups and virtual fibring","authors":"Dawid Kielak","doi":"10.1090/jams/936","DOIUrl":"https://doi.org/10.1090/jams/936","url":null,"abstract":"We show that a finitely generated residually finite rationally solvable (or RFRS) group $G$ is virtually fibred, in the sense that it admits a virtual surjection to $mathbb{Z}$ with a finitely generated kernel, if and only if the first $L^2$-Betti number of $G$ vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of $3$-manifolds.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79054108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-30DOI: 10.1142/s1664360721500016
W. Guo, D. Revin, E. Vdovin
Let $mathfrak{X}$ be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if $A$ is a normal subgroup of a finite group $G$ then the image of an $mathfrak{X}$-maximal subgroup $H$ of $G$ in $G/A$ is not, in general, $mathfrak{X}$-maximal in $G/A$. We say that the reduction $mathfrak{X}$-theorem holds for a finite group $A$ if, for every finite group $G$ that is an extension of $A$ (i. e. contains $A$ as a normal subgroup), the number of conjugacy classes of $mathfrak{X}$-maximal subgroups in $G$ and $G/A$ is the same. The reduction $mathfrak{X}$-theorem for $A$ implies that $HA/A$ is $mathfrak{X}$-maximal in $G/A$ for every extension $G$ of $A$ and every $mathfrak{X}$-maximal subgroup $H$ of $G$. In this paper, we prove that the reduction $mathfrak{X}$-theorem holds for $A$ if and only if all $mathfrak{X}$-maximal subgroups are conjugate in $A$ and classify the finite groups with this property in terms of composition factors.
设$mathfrak{X}$是一类闭于取子群、同态象和扩展的有限群。已知如果$A$是有限群$G$的正规子群，则$G$在$G/A$中的$mathfrak{X}$-极大子群$H$的像一般不是$G/A$中的$mathfrak{X}$-极大子群$H$。我们说$mathfrak{X}$-定理对于有限群$ a $成立，如果对于$ a $的扩展(即包含$ a $作为正规子群)的每一个有限群$G$， $G$和$G/ a $中$mathfrak{X}$-极大子群的共轭类的个数相同。$A$的约简$ mathfrak{X}$定理表明$HA/A$对于$A$的每一个扩展$G$和$G$的每一个$mathfrak{X}$极大子群$H$，在$G/A$中是$mathfrak{X}$最大的。本文证明了$mathfrak{X}$-约简定理对$A$成立当且仅当$mathfrak{X}$-极大子群在$A$中是共轭的，并根据组合因子对具有此性质的有限群进行了分类。
{"title":"The reduction theorem for relatively maximal subgroups","authors":"W. Guo, D. Revin, E. Vdovin","doi":"10.1142/s1664360721500016","DOIUrl":"https://doi.org/10.1142/s1664360721500016","url":null,"abstract":"Let $mathfrak{X}$ be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if $A$ is a normal subgroup of a finite group $G$ then the image of an $mathfrak{X}$-maximal subgroup $H$ of $G$ in $G/A$ is not, in general, $mathfrak{X}$-maximal in $G/A$. We say that the reduction $mathfrak{X}$-theorem holds for a finite group $A$ if, for every finite group $G$ that is an extension of $A$ (i. e. contains $A$ as a normal subgroup), the number of conjugacy classes of $mathfrak{X}$-maximal subgroups in $G$ and $G/A$ is the same. The reduction $mathfrak{X}$-theorem for $A$ implies that $HA/A$ is $mathfrak{X}$-maximal in $G/A$ for every extension $G$ of $A$ and every $mathfrak{X}$-maximal subgroup $H$ of $G$. In this paper, we prove that the reduction $mathfrak{X}$-theorem holds for $A$ if and only if all $mathfrak{X}$-maximal subgroups are conjugate in $A$ and classify the finite groups with this property in terms of composition factors.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82322175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let VB_n be the virtual braid group on n strands and let S_n be the symmetric group on n letters. Let n, m ∈ N such that n ≥ 5, m ≥ 2 and n ≥ m. We determine all possible homomorphisms from VB_n to S_m , from S_n to VB_m and from VB_n to VB_m. As corollaries we get that Out(VB_n) is isomorphic to the Klein group and that VB_n is both Hopfian and co-Hofpian.
{"title":"Virtual braids and permutations","authors":"P. Bellingeri, L. Paris","doi":"10.5802/aif.3336","DOIUrl":"https://doi.org/10.5802/aif.3336","url":null,"abstract":"Let VB_n be the virtual braid group on n strands and let S_n be the symmetric group on n letters. Let n, m ∈ N such that n ≥ 5, m ≥ 2 and n ≥ m. We determine all possible homomorphisms from VB_n to S_m , from S_n to VB_m and from VB_n to VB_m. As corollaries we get that Out(VB_n) is isomorphic to the Klein group and that VB_n is both Hopfian and co-Hofpian.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79499528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Fixing a subgroup $Gamma$ in a group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Delta$ of $G$ with $[Gamma: Gamma cap Delta][Delta : Gamma cap Delta] = n$. For pairs $Gamma leq A$, where $A$ is the automorphism group of a $p$-regular tree and $Gamma$ is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals. For almost all known branch groups $Gamma$ (the first Grigorchuk group, the twisted twin Grigorchuk group, Pervova groups, Gupta-Sidki groups, etc.) acting on $p$-regular trees, this function is precisely $aleph_0$ for any $n = p^k$.
在一个组$G$中固定一个子组$Gamma$，可通约性增长函数将$G$的子组集合$Delta$的基数与$[Gamma: Gamma cap Delta][Delta : Gamma cap Delta] = n$分配给每个$n$。对于对$Gamma leq A$，其中$A$是$p$ -正则树的自同构群，并且$Gamma$是有限生成的，我们证明该函数可以采用有限的、可数的或不可数的基数。对于几乎所有已知的分支群$Gamma$(第一个Grigorchuk群，扭曲双胞胎Grigorchuk群，Pervova群，Gupta-Sidki群，等等)作用于$p$正则树，这个函数对于任何$n = p^k$都是$aleph_0$。
{"title":"Commensurability growth of branch groups","authors":"K. Bou-Rabee, Rachel Skipper, Daniel Studenmund","doi":"10.2140/pjm.2020.304.43","DOIUrl":"https://doi.org/10.2140/pjm.2020.304.43","url":null,"abstract":"Fixing a subgroup $Gamma$ in a group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Delta$ of $G$ with $[Gamma: Gamma cap Delta][Delta : Gamma cap Delta] = n$. For pairs $Gamma leq A$, where $A$ is the automorphism group of a $p$-regular tree and $Gamma$ is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals. For almost all known branch groups $Gamma$ (the first Grigorchuk group, the twisted twin Grigorchuk group, Pervova groups, Gupta-Sidki groups, etc.) acting on $p$-regular trees, this function is precisely $aleph_0$ for any $n = p^k$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91442105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-08-03DOI: 10.1512/IUMJ.2021.70.8257
V. Popov, Y. Zarhin
We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $mathcal L(K)$ generated by ${rm Aut} (K)$ and multiplications by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are isomorphic to a subgroup of $mathcal L(K)$ for a number field $K$ of degree $n$ over $mathbb Q$.
{"title":"Root systems in number fields","authors":"V. Popov, Y. Zarhin","doi":"10.1512/IUMJ.2021.70.8257","DOIUrl":"https://doi.org/10.1512/IUMJ.2021.70.8257","url":null,"abstract":"We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $mathcal L(K)$ generated by ${rm Aut} (K)$ and multiplications by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are isomorphic to a subgroup of $mathcal L(K)$ for a number field $K$ of degree $n$ over $mathbb Q$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81208141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-07-26DOI: 10.2140/agt.2020.20.2885
Timm von Puttkamer, Xiaolei Wu
For a group $G$ we consider the classifying space $E_{mathcal{VC}yc}(G)$ for the family of virtually cyclic subgroups. We show that an Artin group admits a finite model for $E_{mathcal{VC}yc}(G)$ if and only if it is virtually cyclic. This solves a conjecture of Juan-Pineda and Leary and a question of L"uck-Reich-Rognes-Varisco for Artin groups. We then study the conjugacy growth of CAT(0) groups and show that if a CAT(0) group contains a free abelian group of rank two, its conjugacy growth is strictly faster than linear. This also yields an alternative proof for the fact that a CAT(0) cube group admits a finite model for $E_{mathcal{VC}yc}(G)$ if and only if it is virtually cyclic. Our last result deals with the homotopy type of the quotient space $B_{mathcal{VC}yc}(G) = E_{mathcal{VC}yc}(G)/G$. We show for a poly-$mathbb Z$-group $G$, that $B_{mathcal{VC}yc}(G)$ is homotopy equivalent to a finite CW-complex if and only if $G$ is cyclic.
{"title":"Some results related to finiteness properties of groups for families of subgroups","authors":"Timm von Puttkamer, Xiaolei Wu","doi":"10.2140/agt.2020.20.2885","DOIUrl":"https://doi.org/10.2140/agt.2020.20.2885","url":null,"abstract":"For a group $G$ we consider the classifying space $E_{mathcal{VC}yc}(G)$ for the family of virtually cyclic subgroups. We show that an Artin group admits a finite model for $E_{mathcal{VC}yc}(G)$ if and only if it is virtually cyclic. This solves a conjecture of Juan-Pineda and Leary and a question of L\"uck-Reich-Rognes-Varisco for Artin groups. We then study the conjugacy growth of CAT(0) groups and show that if a CAT(0) group contains a free abelian group of rank two, its conjugacy growth is strictly faster than linear. This also yields an alternative proof for the fact that a CAT(0) cube group admits a finite model for $E_{mathcal{VC}yc}(G)$ if and only if it is virtually cyclic. Our last result deals with the homotopy type of the quotient space $B_{mathcal{VC}yc}(G) = E_{mathcal{VC}yc}(G)/G$. We show for a poly-$mathbb Z$-group $G$, that $B_{mathcal{VC}yc}(G)$ is homotopy equivalent to a finite CW-complex if and only if $G$ is cyclic.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83418320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $G$ be a group. A subset $F subset G$ is called irreducibly faithful if there exists an irreducible unitary representation $pi$ of $G$ such that $pi(x) ne text{id}$ for all $x in F smallsetminus {e}$. Otherwise $F$ is called irreducibly unfaithful. Given a positive integer $n$, we say that $G$ has Property $P(n)$ if every subset of size $n$ is irreducibly faithful. Every group has $P(1)$, by a classical result of Gelfand and Raikov. Walter proved that every group has $P(2)$. It is easy to see that some groups do not have $P(3)$. We provide a complete description of the irreducibly unfaithful subsets of size $n$ in a (finite or infinite) countable group $G$ with Property $P(n-1)$: it turns out that such a subset is contained in a finite elementary abelian normal subgroup of $G$ of a particular kind. We deduce a characterization of Property $P(n)$ purely in terms of the group structure. It follows that, if a countable group $G$ has $P(n-1)$ and does not have $P(n)$, then $n$ is the cardinality of a projective space over a finite field. A group $G$ has Property $Q(n)$ if, for every subset $F subset G$ of size at most $n$, there exists an irreducible unitary representation $pi$ of $G$ such that $pi(x) ne pi(y)$ for any distinct $x, y$ in $F$. Every group has $Q(2)$. For countable groups, it is shown that Property $Q(3)$ is equivalent to $P(3)$, Property $Q(4)$ to $P(6)$, and Property $Q(5)$ to $P(9)$. For $m, n ge 4$, the relation between Properties $P(m)$ and $Q(n)$ is closely related to a well-documented open problem in additive combinatorics.
让$G$成为一个团体。一个子集$F subset G$被称为不可约忠实的，如果存在一个不可约的酉表示$pi$的$G$，使得$pi(x) ne text{id}$对于所有$x in F smallsetminus {e}$。否则$F$被称为不可还原的不忠。给定一个正整数$n$，如果大小为$n$的每个子集都是不可约忠实的，我们说$G$具有属性$P(n)$。根据Gelfand和Raikov的经典结果，每个群体都有$P(1)$。Walter证明了每个组都有$P(2)$。很容易看出，有些组没有$P(3)$。给出了具有$P(n-1)$性质的(有限或无限)可数群$G$中大小为$n$的不可约不忠实子集的完整描述，证明了这样的子集包含在特定种类的有限初等阿贝尔正规子群$G$中。我们纯粹从群体结构的角度推导出属性$P(n)$的特征。由此可知，如果可数群$G$有$P(n-1)$而不有$P(n)$，则$n$是有限域上射影空间的基数。一个群$G$具有$Q(n)$的属性，如果对于每个不超过$n$大小的子集$F subset G$，存在$G$的一个不可约的幺正表示$pi$，使得$pi(x) ne pi(y)$对于$F$中的任意一个不同的$x, y$。每个组都有$Q(2)$。对于可数群，可以看出Property $Q(3)$等价于$P(3)$, Property $Q(4)$等价于$P(6)$, Property $Q(5)$等价于$P(9)$。对于$m, n ge 4$，属性$P(m)$和$Q(n)$之间的关系与加性组合学中一个记录良好的开放问题密切相关。
{"title":"Groups with irreducibly unfaithful subsets for unitary representations","authors":"P. Caprace, P. Harpe","doi":"10.5802/CML.61","DOIUrl":"https://doi.org/10.5802/CML.61","url":null,"abstract":"Let $G$ be a group. A subset $F subset G$ is called irreducibly faithful if there exists an irreducible unitary representation $pi$ of $G$ such that $pi(x) ne text{id}$ for all $x in F smallsetminus {e}$. Otherwise $F$ is called irreducibly unfaithful. Given a positive integer $n$, we say that $G$ has Property $P(n)$ if every subset of size $n$ is irreducibly faithful. Every group has $P(1)$, by a classical result of Gelfand and Raikov. Walter proved that every group has $P(2)$. It is easy to see that some groups do not have $P(3)$. \u0000We provide a complete description of the irreducibly unfaithful subsets of size $n$ in a (finite or infinite) countable group $G$ with Property $P(n-1)$: it turns out that such a subset is contained in a finite elementary abelian normal subgroup of $G$ of a particular kind. We deduce a characterization of Property $P(n)$ purely in terms of the group structure. It follows that, if a countable group $G$ has $P(n-1)$ and does not have $P(n)$, then $n$ is the cardinality of a projective space over a finite field. \u0000A group $G$ has Property $Q(n)$ if, for every subset $F subset G$ of size at most $n$, there exists an irreducible unitary representation $pi$ of $G$ such that $pi(x) ne pi(y)$ for any distinct $x, y$ in $F$. Every group has $Q(2)$. For countable groups, it is shown that Property $Q(3)$ is equivalent to $P(3)$, Property $Q(4)$ to $P(6)$, and Property $Q(5)$ to $P(9)$. For $m, n ge 4$, the relation between Properties $P(m)$ and $Q(n)$ is closely related to a well-documented open problem in additive combinatorics.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90523499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems.
{"title":"Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks","authors":"S. Francaviglia, A. Martino","doi":"10.1090/tran/8333","DOIUrl":"https://doi.org/10.1090/tran/8333","url":null,"abstract":"This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. \u0000In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90984786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is the second of two papers in which we investigate the properties of displacement functions of automorphisms of free groups (more generally, free products) on the Culler-Vogtmann Outer space $CV_n$ and its simplicial bordification. We develop a theory for both reducible and irreducible autormorphisms. As we reach the bordification of $CV_n$ we have to deal with general deformation spaces, for this reason we developed the theory in such generality. In first paper~cite{FMpartI} we studied general properties of the displacement functions, such as well-orderability of the spectrum and the topological characterization of min-points via partial train tracks (possibly at infinity). This paper is devoted to proving that for any automorphism (reducible or not) any level set of the displacement function is connected. As an application, this result provides a stopping procedure for brute force search algorithms in $CV_n$. We use this to reprove two known algorithmic results: the conjugacy problem for irreducible automorphisms and detecting irreducibility of automorphisms. Note: the two papers were originally packed together in the preprint arxiv:1703.09945. We decided to split that paper following the recommendations of a referee.
{"title":"Displacements of automorphisms of free groups II: Connectivity of level sets and decision problems","authors":"S. Francaviglia, A. Martino","doi":"10.1090/tran/8535","DOIUrl":"https://doi.org/10.1090/tran/8535","url":null,"abstract":"This is the second of two papers in which we investigate the properties of displacement functions of automorphisms of free groups (more generally, free products) on the Culler-Vogtmann Outer space $CV_n$ and its simplicial bordification. We develop a theory for both reducible and irreducible autormorphisms. As we reach the bordification of $CV_n$ we have to deal with general deformation spaces, for this reason we developed the theory in such generality. In first paper~cite{FMpartI} we studied general properties of the displacement functions, such as well-orderability of the spectrum and the topological characterization of min-points via partial train tracks (possibly at infinity). This paper is devoted to proving that for any automorphism (reducible or not) any level set of the displacement function is connected. As an application, this result provides a stopping procedure for brute force search algorithms in $CV_n$. We use this to reprove two known algorithmic results: the conjugacy problem for irreducible automorphisms and detecting irreducibility of automorphisms. Note: the two papers were originally packed together in the preprint arxiv:1703.09945. We decided to split that paper following the recommendations of a referee.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88050586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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