We discuss the following question of G. Makanin from ``Kourovka notebook'': does there exist an algorithm to determine is for an arbitrary pair of words $U$ and $V$ of a free group $F_n$ and an arbitrary automorphism $ϕ\in Aut(F_n)$ the equation $ϕ(X) U = V X$ solvable in $F_n$? We give the affirmative answer in the case when an automorphism is virtually inner, i.e. some its non-zero power is an inner automorphism of $F_n$.
We prove a conjecture due to Makanin: if a and b are elements of the Artin braid group B_n such that a^k=b^k for some nonzero integer k, then a and b are conjugate. The proof involves the Nielsen-Thurston classification of braids.
We study twisted conjugacy classes of the unit element in different groups. Fel'shtyn and Troitsky showed that the twisted conjugacy class of the unit element of an abelian group is a subgroup for every automorphism. The structure is investigated of a group whose twisted conjugacy class of the unit element is a subgroup for every automorphism (inner automorphism).
This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a precise calculation of conjugacy…
Let $f$ be an automorphism of a group $G$. Two elements $x, y$ in $G$ are said to be in the same $f$-twisted conjugacy class if there exists an element $z$ in $G$ such that $y=z x f(z^{-1})$. This is an equivalence relation known as $f$-twisted conjugacy. If the number $R(f)$ of $f$-twisted conjugacy classes is infinite for every automorphism $f$ of $G$ one says that $G$ has the $R_\infty$-property. We show that the Richard Thompson group $T$ has the $R_\infty$-property.
Let $ϕ:G\to G$ be an automorphism of a group which is a free-product of finitely many groups each of which is freely indecomposable and two of the factors contain proper finite index characteristic subgroups. We show that $G$ has infinitely many $ϕ$-twisted conjugacy classes. As an application, we show that if $G$ is the fundamental group of a three-manifold that is not irreducible, then $G$ has property $R_\infty$, that is, there are infinitely many $ϕ$-twisted conjugacy classes in $G$ for every automorphism $ϕ$ of $G$.
In this note we solve the twisted conjugacy problem for braid groups, i.e. we propose an algorithm which, given two braids $u,v\in B_n$ and an automorphism $ϕ\in Aut (B_n)$, decides whether $v=(ϕ(x))^{-1}ux$ for some $x\in B_n$. As a corollary, we deduce that each group of the form $B_n \rtimes H$, a semidirect product of the braid group $B_n$ by a torsion-free hyperbolic group $H$, has solvable conjugacy problem.
We prove that Chevalley group over the field $F$ of zero characteristic possess $R_{\infty}$ property, if $F$ has torsion group of automorphisms or $F$ is an algebraically closed field which has finite transcendence degree over $\mathbb{Q}$. As a consequence we obtain that the twisted conjugacy class $[e]_φ$ of unit element is a subgroup of Chevalley group if and only if $φ$ is central automorphism.
A group $G$ has property $R_\infty$ if for every $ϕ\in Aut(G)$, there are an infinite number of $ϕ$-twisted conjugacy classes of elements in $G$. In this note, we determine the $R_\infty$-property for $G=π_1(M)$ for all geometric $3$-manifolds $M$.
We say that $x,y\in Γ$ are in the same $ϕ$-twisted conjugacy class and write $x\sim_ϕy$ if there exists an element $γ\in Γ$ such that $y=γxϕ(γ^{-1})$. This is an equivalence relation on $Γ$ called the $ϕ$-twisted conjugacy. Let $R(ϕ)$ denote the number of $ϕ$-twisted conjugacy classes in $Γ$. If $R(ϕ)$ is infinite for all $ϕ\in Aut(Γ)$, we say that $Γ$ has the $R_\infty$-property. The purpose of this note is to show that the symmetric group $S_\infty$, the Houghton groups and the pure symmetric automorphism groups have the $R_\infty$-property. We show, also, that the Richard Thompson group $T$…
Let $G$ be a non-compact semisimple Lie group with finite centre and finitely many components. We show that any finitely generated group $Γ$ which is quasi-isometric to an irreducible lattice in $G$ has the $R_\infty$-property, namely, that there are infinitely $ϕ$-twisted conjugacy classes for every automorphism $ϕ$ of $Γ$. Also, we show that any lattice in $G$ has the $R_\infty$-property, extending our earlier result for irreducible lattices.
In this paper we study twisted conjugacy classes and the $R_{\infty}$-property for classical linear groups. In particular, we prove that the general linear group ${\rm GL}_n(K)$ and the special linear group ${\rm SL}_n(K)$ possess $R_{\infty}$-property, if $n\geq3$ and $K$ is an infinite integral domain with trivial group of automorphisms (Theorem 1), or $K$ is an integral domain, which has a zero characteristic and for which ${\rm Aut}(K)$ is torsion (Theorem 2).
Let $k$ be an algebraically closed field, $G$ a linear algebraic group over $k$ and $φ\in Aut(G)$, the group of all algebraic group automorphisms of $G$. Two elements $x, y$ of $G$ are said to be $φ$-twisted conjugate if $y=gxφ(g)^{-1}$ for some $g\in G$. In this paper we prove that for a connected non-solvable linear algebraic group $G$ over $k$, the number of its $φ$-twisted conjugacy classes is infinite for every $φ\in Aut(G)$.
If $ϕ$ is an automorphism of a group $G$ and $x,y\in G$, we say that $x$ and $y$ are $ϕ$-twisted conjugates if there exists an $z\in G$ such that $y=z.x.ϕ(z^{-1})$. This is an equivalence relation. If there are infinitely many $ϕ$-twisted conjugacy classes for every automorphism $ϕ$ of $G$ we say that $G$ has the $R_\infty$-property. We prove that the generalized Richard Thompson groups $F_n$ and $F(l,A,P)$ have the $R_\infty$-property.
This thesis deals with the conjugacy problem in groups and its twisted variants. We analyze recent results by Bogopolski, Martino, Maslakova and Ventura on the twisted conjugacy problem in free groups and its implication for the conjugacy problem in free-by-cyclic groups and some further group extensions. We also consider the doubly-twisted conjugacy problem in free groups. Staecker has developed an algorithm for deciding doubly-twisted conjugacy relations in the case where the involved homomorphisms satisfy a certain remnant inequality. We show how a similar condition affects the equalizer su…
We give a geometric proof based on recent work of Eskin, Fisher and Whyte that the lamplighter group $L_n$ has infinitely many twisted conjugacy classes for any automorphism $\vp$ only when $n$ is divisible by 2 or 3, originally proved by Gonçalves and Wong. We determine when the wreath product $G \wr \Z$ has this same property for several classes of finite groups $G$, including symmetric groups and some nilpotent groups.
A group $G$ is said to have property $R_{\infty}$ if for every automorphism $φ\in {\rm Aut}(G)$, the cardinality of the set of $φ$-twisted conjugacy classes is infinite. Many classes of groups are known to have such property. However, very few examples are known for which $R_{\infty}$ is {\it geometric}, i.e., if $G$ has property $R_{\infty}$ then any group quasi-isometric to $G$ also has property $R_{\infty}$. In this paper, we give examples of groups and conditions under which $R_{\infty}$ is preserved under commensurability. The main tool is to employ the Bieri-Neumann-Strebel invariant.
Let $G$ be a group and $φ$ be an automorphism of $G$. Two elements $x, y$ of $G$ are said to be $φ$-twisted conjugate if $y=gxφ(g)^{-1}$ for some $g\in G$. A group $G$ has the $R_{\infty}$-property if the number of $φ$-twisted conjugacy classes is infinite for every automorphism $φ$ of $G$. In this paper we prove that the big mapping class group $MCG(S)$ possesses the $R_{\infty}$-property under some suitable conditions on the infinite-type surface $S$. As an application we also prove that the big mapping class group possesses the $R_\infty$-property if and only if it satisfies the $S_{\infty}…
Let $N$ be a finitely generated nilpotent group. Algorithm is constructed such, that for every automorphism $ϕ\in Aut(N)$ defines the Reidemeister number $R(ϕ).$ It is proved that any free nilpotent group of rank $r = 2$ or $r = 3$ and class $c \geq 4r,$ or rank $r \geq 4$ and class $c \geq 2r,$ belongs to the class $R_{\infty}.$
For a fixed $n\ge2$, the Houghton group $H_n$ consists of bijections of $X_n=\{1,\ldots,n\} \times \mathbb{N}$ that are `eventually translations' of each copy of $\mathbb{N}$. The Houghton groups have been shown to have solvable conjugacy problem. In general solvable conjugacy problem does not imply that all finite extensions and finite index subgroups have solvable conjugacy problem. Our main theorem is that a stronger result holds: for any $n\ge2$ and any group $G$ commensurable to $H_n$, $G$ has solvable conjugacy problem.
Let $F$ be a subfield of the algebraic closure of a finite field $\mathbb{F}_p$, $p \ne 2$, and let $R$ denote any ring such that $F[t] \subset R \subsetneq F(t)$. Let $G$ be a classical Chevalley group of adjoint type defined over $R$. We prove that the group $G(R)$ has the $R_{\infty}$-property.
We study groups $G$ where the $φ$-conjugacy class $[e]_φ=\{g^{-1}φ(g)~|~g\in G\}$ of the unit element is a subgroup of $G$ for every automorphism $φ$ of $G$. If $G$ has $n$ generators, then we prove that the $k$-th member of the lower central series has a finite verbal width bounded in terms of $n,k$. Moreover, we prove that if such group $G$ satisfies the descending chain condition for normal subgroups, then $G$ is nilpotent. Finally, if $G$ is a finite abelian-by-cyclic group, we construct a good upper bound of the nilpotency class of $G$.
There currently exists no algebraic algorithm for computing twisted conjugacy classes in free groups. We propose a new technique for deciding twisted conjugacy relations using nilpotent quotients. Our technique is generalization of the common abelianization method, but admits significantly greater rates of success. We present experimental results demonstrating the efficacy of the technique, and detail how it can be applied the related settings of surface groups and doubly twisted conjugacy.
An algorithm is constructed that, when given an explicit presentation of a finitely generated nilpotent group $G,$ decides for any pair of endomorphisms $φ, ψ: G \to G$ and any pair of elements $u, v \in G,$ whether or not the equation $(xφ)u = v (xψ)$ has a solution $x \in G.$ Thus it is shown that the problem of the title is decidable. Also we present an algorithm that produces a finite set of generators of the subgroup (equalizer) $Eq_{φ, ψ}(G) \leq G$ of all elements $u \in G$ such that $u φ= u ψ.$
In this short article, we prove that any automorphism of the R. Thompson's group $F$ has infinitely many twisted conjugacy classes. The result follows from the work of Matthew Brin, together with a standard facts on R. Thompson's group $F$, and elementary properties of the Reidemeister numbers.
Let $R$ be an integral domain of zero characteristic. In this note we study the Reidemeister spectrum of the group ${\rm UT}_n(R)$ of unitriangular matrices over $R$. We prove that if $R^+$ is finitely generated and $n>2|R^*|$, then ${\rm UT}_n(R)$ possesses the $R_{\infty}$-property, i. e. the Reidemeister spectrum of ${\rm UT}_n(R)$ contains only $\infty$, however, if $n\leq|R^*|$, then the Reidemeister spectrum of ${\rm UT}_n(R)$ has nonempty intersection with $\mathbb{N}$. If $R$ is a field, then we prove that the Reidemeister spectrum of ${\rm UT}_n(R)$ coincides with $\{1,\infty\}$, i. e…
Given a group automorphism $ϕ:Γ\to Γ$, one has an action of $Γ$ on itself by $ϕ$-twisted conjugacy, namely, $g.x=gxϕ(g^{-1})$. The orbits of this action are called $ϕ$-conjugacy classes. One says that $Γ$ has the $R_\infty$-property if there are infinitely many $ϕ$-conjugacy classes for every automorphism $ϕ$ of $Γ$. In this paper we show that any irreducible lattice in a connected semi simple Lie group having finite centre and rank at least 2 has the $R_\infty$-property.
Let G be a simple algebraic group over an algebraically closed field of good odd characteristic, and let theta be an automorphism of G arising from an involution of its Dynkin diagram. We show that the spherical theta-twisted conjugacy classes are precisely those intersecting only Bruhat cells corresponding to twisted involutions in the Weyl group. We show how the analogue of this statement fails in the triality case. We generalize to good odd characteristic J-H. Lu's dimension formula for spherical twisted conjugacy classes.
We study the notion of twisted conjugacy separability (essentially introduced in our previous paper for a proof of twisted version of Burnside-Frobenius theorem) and some related properties. We give examples of groups with and without this property and study its behavior under some extensions. An affirmative answer to the twisted Dehn conjugacy problem for polycyclic-by-finite group is obtained. Some problems for the further study are indicated.
We construct two practical algorithms for twisted conjugacy classes of polycyclic-by-finite groups. The first algorithm determines whether two elements of a group are twisted conjugate for two given endomorphisms, under the condition that the Reidemeister coincidence number of these endomorphisms is finite. The second algorithm determines representatives of the Reidemeister coincidence classes of two endomorphisms if their Reidemeister coincidence number is finite, or returns "fail" if the Reidemeister coincidence number is infinite.
We give an easily checkable algebraic condition which implies that two elements of a finitely generated free group are members of distinct doubly-twisted conjugacy classes with respect to a pair of homomorphisms. We further show that this criterion is satisfied with probability 1 when the homomorphisms and elements are chosen at random.
We give two results for computing doubly-twisted conjugacy relations in free groups with respect to homomorphisms $ϕ$ and $ψ$ such that certain remnant words from $ϕ$ are longer than the images of generators under $ψ$. Our first result is a remnant inequality condition which implies that two words $u$ and $v$ are not doubly-twisted conjugate. Further we show that if $ψ$ is given and $ϕ$, $u$, and $v$ are chosen at random, then the probability that $u$ and $v$ are not doubly-twisted conjugate is 1. In the particular case of singly-twisted conjugacy, this means that if $ϕ$, $u$, and $v$ are chos…
Let $G$ be a linear algebraic group over an algebraically closed field $k$ and $\mathrm{Aut}_{\mathrm{alg}}(G)$ the group of all algebraic group automorphisms of $G$. For every $φ\in \mathrm{Aut}_{\mathrm{alg}}(G)$ let $\mathcal{R}(φ)$ denote the set of all orbits of the $φ$-twisted conjugacy action of $G$ on itself (given by $(g,x)\mapsto gxφ(g^{-1})$, for all $g,x\in G$). We say that $G$ has the algebraic $R_\infty$-property if $\mathcal{R}(φ)$ is infinite for every $φ\in \mathrm{Aut}_{\mathrm{alg}}(G)$. In \citep{bb} we have shown that this property is satisfied by every connected non-solva…
Given an automorphism $ϕ:Γ\to Γ$, one has an action of $Γ$ on itself by $ϕ$-twisted conjugacy, namely, $g.x=gxϕ(g^{-1})$. The orbits of this action are called $ϕ$-twisted conjugacy classes. One says that $Γ$ has the $R_\infty$-property if there are infinitely many $ϕ$-twisted conjugacy classes for every automorphism $ϕ$ of $Γ$. In this paper we show that SL$(n,\mathbb{Z})$ and its congruence subgroups have the $R_\infty$-property. Further we show that any (countable) abelian extension of $Γ$ has the $R_\infty$-property where $Γ$ is a torsion free non-elementary hyperbolic group, or SL$(n,\math…
Let $ϕ:G\to G$ be an automorphism of an infinite group $G$. One has an equivalence relation $\sim_ϕ$ on $G$ defined as $x\sim_ϕy$ if there exists a $z\in G$ such that $y=zxϕ(z^{-1})$. The equivalence classes are called $ϕ$-twisted conjugacy classes and the set $G/\!\!\sim_ϕ$ of equivalence classes is denoted $\mathcal R(ϕ)$. The cardinality $R(ϕ)$ of $\mathcal R(ϕ)$ is called the Reidemeister number of $ϕ$. We write $R(ϕ)=\infty$ when $\mathcal R(ϕ)$ is infinite. We say that $G$ has the $R_\infty$-{\it property} if $R(ϕ)=\infty$ for every automorphism $ϕ$ of $G$. We show that the groups $G=GL_…
Let $G$ be a finitely generated polyfree group. If $G$ has nonzero Euler characteristic then we show that $Aut(G)$ has a finite index subgroup in which every automorphism has infinite Reidemeister number. For certain $G$ of length 2, we show that the number of Reidemeister classes of every automorphism is infinite.
Given a short exact sequence of groups with certain conditions, $1\to F\to G\to H\to 1$, we prove that $G$ has solvable conjugacy problem if and only if the corresponding action subgroup $A\leqslant Aut(F)$ is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form $\mathbb{Z}^2\rtimes F_m$, $F_2\rtimes F_m$, $F_n \rtimes \mathbb{Z}$, and $\mathbb{Z}^n \rtimes_A F_m$ with virtually solvable action group $A\leqslant GL_n(\mathbb{Z})$. Also, we give an easy way of constructing groups of the form $\mathbb{Z}^4\rtimes F_n$ and $F_3\rti…
We say that a group has property $R_{\infty}$ if any group automorphism has an infinite number of twisted conjugacy classes. Fel'shtyn and Goncalves prove that the solvable Baumslag-Solitar groups BS(1,m) have property $R_{\infty}$. We define a solvable generalization $Γ(S)$ of these groups which we show to have property $R_{\infty}$. We then show that property $R_{\infty}$ is geometric for these groups, that is, any group quasi-isometric to $Γ(S)$ has property $R_{\infty}$ as well.
We establish a connection between the generalized conjugacy problem for a $G$-by-$\mathbb{Z}$ group, $GCP(G \rtimes \mathbb{Z})$, and two algorithmic problems for $G$: the generalized Brinkmann's conjugacy problem, $GBrCP(G)$, and the generalized twisted conjugacy problem, $GTCP(G)$. We explore this connection for generalizations of different kinds: relative to finitely generated subgroups, to theirs cosets, or to recognizable, rational, context-free or algebraic subsets of the group. Using this result, we are able to prove that $GBrCP(G)$ is decidable (with respect to cosets) when $G$ is a vi…
We prove for residually finite groups the following long standing conjecture: the number of twisted conjugacy classes of an automorphism of a finitely generated group is equal (if it is finite) to the number of finite dimensional irreducible unitary representations being invariant for the dual of this automorphism. Also, we prove that any finitely generated residually finite non-amenable group has the R-infinity property (any automorphism has infinitely many twisted conjugacy classes). This gives a lot of new examples and covers many known classes of such groups.
A generalized Baumslag-Solitar group is the fundamental group of a graph of groups all of whose vertex and edge groups are infinite cyclic. Levitt proves that any generalized Baumslag-Solitar group has property R-infinity, that is, any automorphism has an infinite number of twisted conjugacy classes. We show that any group quasi-isometric to a generalized Baumslag-Solitar group also has property R-infinity. This extends work of the authors proving that any group quasi-isometric to a solvable Baumslag-Solitar BS(1,n) group has property R-infinity, and relies on the classification of generalized…
We prove that the symplectic group $Sp(2n,\mathbb Z)$ and the mapping class group $Mod_{S}$ of a compact surface $S$ satisfy the $R_{\infty}$ property. We also show that $B_n(S)$, the full braid group on $n$-strings of a surface $S$, satisfies the $R_{\infty}$ property in the cases where $S$ is either the compact disk $D$, or the sphere $S^2$. This means that for any automorphism $ϕ$ of $G$, where $G$ is one of the above groups, the number of twisted $ϕ$-conjugacy classes is infinite.
We calculate asymptotic estimates for the conjugacy growth function of finitely generated class 2 nilpotent groups whose derived subgroup is infinite cyclic, including the so-called higher Heisenberg groups. We prove that these asymptotics are stable when passing to commensurable groups, by understanding their twisted conjugacy growth. We also use these estimates to prove that, in certain cases, the conjugacy growth series cannot be a holonomic function.
