Twisted Conjugacy Research Articles

Twisted conjugacy in PL-homeomorphism groups of the circle

Given an automorphism $$\phi :\Gamma \rightarrow \Gamma $$ of a group, one has a left action of $$\Gamma $$ on itself defined as $$g.x=gx\phi (g^{-1})$$ . The orbits of this action are called the Reidemeister classes or $$\phi $$ -twisted conjugacy classes. We denote by $$R(\phi )\in {\mathbb {N}}\cup \{\infty \}$$ the Reidemeister number of $$\phi $$ , namely, the cardinality of the orbit space $${\mathcal {R}}(\phi )$$ if it is finite and $$R(\phi )=\infty $$ if $${\mathcal {R}}(\phi )$$ is infinite. The group $$\Gamma $$ is said to have the $$R_\infty $$ -property if $$R(\phi )=\infty $$ for all automorphisms $$\phi \in {\text {Aut}}(\Gamma )$$ . We show that the generalized Thompson group T(r, A, P) has the $$R_\infty $$ -property when the slope group $$P\subset {\mathbb {R}}^\times _{>0}$$ is not cyclic.

Effective twisted conjugacy separability of nilpotent groups

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 possibly sharp upper bound for the conjugacy separability for finitely generated nilpotent groups of nilpotency class 2.

Twisted conjugacy in Houghton's groups

For a fixed n∈{2,3,…}, the Houghton group Hn consists of bijections of Xn={1,…,n}×N that are ‘eventually translations’ of each copy of 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∈{2,3,…} and any group G commensurable to Hn, G has solvable conjugacy problem.

The conjugacy problem in extensions of Thompson’s group F

We solve the twisted conjugacy problem on Thompson’s group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut+(F) are orbit decidable provided a certain conjecture on Thompson’s group T is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [5], we construct a family of free extensions of F where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak–Fel’shtyn–Goncalves in [4] showing that F has property R∞, and which can be extended to show that Thompson’s group T also has property R∞.

The R ∞ and S ∞ properties for linear algebraic groups

Abstract In this paper we study twisted conjugacy classes and isogredience classes for automorphisms of reductive linear algebraic groups. We show that reductive linear algebraic groups over some fields of zero characteristic possess the R ∞ ${R_{\infty}}$ and S ∞ ${S_{\infty}}$ properties.

The R∞ property for nilpotent quotients of surface groups

A group is said to have the property if, for any automorphism of , the number of twisted conjugacy classes (or Reidemeister classes) is infinite. It is well known that when is the fundamental group of a closed surface of negative Euler characteristic, it has the property. In this work, we compute the least integer , called the ‐nilpotency degree of , such that the group has the property, where is the th term of the lower central series of . We show that for the fundamental group of any orientable closed surface of genus . For the fundamental group of the non‐orientable surface (the connected sum of projective planes) this number is (when ). A similar concept is introduced using the derived series of a group . Namely, the ‐solvability degree of , which is the least integer such that the group has the property. We show that the fundamental group of an orientable closed surface has ‐solvability degree 2. As a by‐product of our research, we find a lot of new examples of nilmanifolds on which every self‐homotopy equivalence can be deformed into a fixed point free map.

Automorphisms of higher rank lamplighter groups

Let [Formula: see text] denote the group whose Cayley graph with respect to a particular generating set is the Diestel–Leader graph [Formula: see text], as described by Bartholdi, Neuhauser and Woess. We compute both [Formula: see text] and [Formula: see text] for [Formula: see text], and apply our results to count twisted conjugacy classes in these groups when [Formula: see text]. Specifically, we show that when [Formula: see text], the groups [Formula: see text] have property [Formula: see text], that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when [Formula: see text] the lamplighter groups [Formula: see text] have property [Formula: see text] if and only if [Formula: see text].

The twisted conjugacy problem for finitely generated free groups

Let G be a finitely generated free group and let φ∈End(G) be an endomorphism of G. In this paper we prove that the twisted conjugacy problem for φ is algorithmically solvable in the special case of φ having remnant. This case covers a significant set of endomorphisms. It is proved in Wagner (1999) [14] that almost all endomorphisms of G have remnant in a sense that can be made precise in terms of probability.For φ∈End(G) having remnant, we provide an upper bound on the length of elements z∈G that need to be checked to solve the twisted conjugate problem for φ so that the algorithm is simple to use for a computer search. Our new algorithm improves on existing algorithms which can only handle homomorphisms with remnant words of length at least 2.

The Post correspondence problem in groups

Abstract We generalize the classical Post correspondence problem (𝐏𝐂𝐏 n ) and its non-homogeneous variation (𝐍𝐏𝐂𝐏 n ) to non-commutative groups and study the computational complexity of these new problems. We observe that 𝐏𝐂𝐏 n is closely related to the equalizer problem in groups, while 𝐍𝐏𝐂𝐏 n is connected to the double twisted conjugacy problem for endomorphisms. Furthermore, it is shown that one of the strongest forms of the word problem in a group G (we call it the hereditary word problem) can be reduced to 𝐍𝐏𝐂𝐏 n in G in polynomial time. The main results are that 𝐏𝐂𝐏 n is decidable in a finitely generated nilpotent group in polynomial time, while 𝐍𝐏𝐂𝐏 n is undecidable in any group containing free non-abelian subgroups (though the argument is very different from the classical case of free semigroups). We show that the double endomorphism twisted conjugacy problem is undecidable in free groups of sufficiently large finite rank. We also consider the bounded 𝐏𝐂𝐏 and observe that it is in 𝐍𝐏 for any group with 𝐏-time decidable word problem, meanwhile it is 𝐍𝐏-hard in any group containing free non-abelian subgroups. In particular, the bounded 𝐏𝐂𝐏 is 𝐍𝐏-complete in non-elementary hyperbolic groups and non-abelian right angle Artin groups.

CARTAN DECOMPOSITION FOR COMPLEX LOOP GROUPS

In this note we formulate and prove a version of Cartan decomposition for holomorphic loop groups, similar to Cartan decomposition for p-adic loop groups, discussed in [3], [6]. The main technical tool that we use is the (well-known) interpretation of twisted conjugacy classes in the holomorphic loop group in terms of principal holomorphic bundles on an elliptic curve.

Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups

AbstractGiven a group automorphismϕ: Γ → Γ, one has an action of Γ on itself byϕ-twisted conjugacy, namely,g.x = gxϕ(g-1). The orbits of this action are calledϕ-twisted conjugacy classes. One says that Γ has theR∞-property if there are infinitely manyϕ-twisted conjugacy classes for every automorphismϕof Γ. In this paper we show that SL(n; Z) and its congruence subgroups have the R8-property. Further we show that any (countable) abelian extension of Γ has the R8-property where Γ is a torsion free non-elementary hyperbolic group, or SL(n; Z); Sp(2n; Z) or a principal congruence subgroup of SL(n; Z) or the fundamental group of a complete Riemannian manifold of constant negative curvature.

Twisted conjugacy in braid groups

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 $\phi \in Aut (B_n)$, decides whether $v=(\phi (x))^{-1}ux$ 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.

Twisted conjugacy classes of the unit element

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).

Elements with finite Coxeter part in an affine Weyl group

Let Wa be an affine Weyl group and η:Wa→W0 be the natural projection to the corresponding finite Weyl group. We say that w∈Wa has finite Coxeter part if η(w) is conjugate to a Coxeter element of W0. The elements with finite Coxeter part are a union of conjugacy classes of Wa. We show that for each conjugacy class O of Wa with finite Coxeter part there exists a unique maximal proper parabolic subgroup WJ of Wa, such that the set of minimal length elements in O is exactly the set of Coxeter elements in WJ. Similar results hold for twisted conjugacy classes.

From conjugacy classes in the Weyl group to unipotent classes, III

Let G G be an affine algebraic group over an algebraically closed field whose identity component G 0 G^{0} is reductive. Let W W be the Weyl group of G G and let D D be a connected component of G G whose image in G / G 0 G/G^{0} is unipotent. In this paper we define a map from the set of “twisted conjugacy classes” in W W to the set of unipotent G 0 G^{0} -conjugacy classes in D D generalizing an earlier construction which applied when G G is connected.

On spherical twisted conjugacy classes

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.

Twisted conjugacy classes in general and special linear groups

We consider twisted conjugacy classes and the R ∞-property for classical linear groups. In particular, it is stated that the general linear group GL n (K) and the special linear group SL n (K), where $$ n \geqslant 3 $$ , possess the R ∞-property if either K is an infinite integral domain with trivial automorphism group or K is an integral domain containing a subring of integers, whose automorphism group Aut(K) is finite. By an integral domain we mean a commutative ring with identity which has no zero divisors.

A new key exchange protocol using two layers of security

In this paper we present a New Key Exchange Protocol with two layers of security. For this purpose, we consider Discrete Heisenberg Group as our platform group. Firstly by considering the Triple Decomposition Problem in Discrete Heisenberg group we arrive at a shared secret key: This forms the first layer of security. Then we make use of this key in forming the Key exchange Protocol using Twisted Conjugacy Search problem implemented in the Discrete Heisenberg group.

The generalized conjugacy problem for virtually free groups

The generalized conjugacy problem (has g a conjugate in K for K rational?) is solved for [f.g. free]-by-finite groups with constraints that go beyond the context-free level, a new result for the free group itself. Moldavanskii’s theorem on simultaneous conjugacy of f.g. subgroups of a free group is also generalized for [f.g. free]-byfinite groups and this wider class of constraints. The solution set of the equation x−1gφ(x) ∈ K in the free group (φ a virtually inner automorphism, K rational) is shown to be rational and effectively constructible, and a similar result is proved for the equation xgx−1 ∈ K in a [f.g. free]-by-finite group. The twisted conjugacy problem with context-free constraints is also proved to be decidable for the free group.

Remnant inequalities and doubly-twisted conjugacy in free groups

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 asymptotic 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 chosen at random, then u and v are not in the same singly-twisted conjugacy class with asymptotic probability 1. Our second result generalizes Kim’s “bounded solution length”. We give an algorithm for deciding doubly-twisted conjugacy relations in the case where φ and ψ satisfy a similar remnant inequality. In the particular case of singly-twisted conjugacy, our algorithm suffices to decide any twisted conjugacy relation if φ has remnant words of length at least 2. As a consequence of our generic properties we give an elementary proof of a recent result of Martino, Turner, and Ventura, that computes the densities of the sets of injective and surjective homomorphisms from one free group to another. We further compute the expected value of the density of the image of a homomorphism.