The conjugacy problem in extensions of Thompson’s group F

Abstract

Abstract. 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 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

“…The problem of determining which classes of discrete infinite groups have the-R ∞ property is an area of active research initiated by Fel'shtyn and Hill in 1994 [9]. Later, it was shown by various authors that the following groups have the R ∞property: non-elementary Gromov hyperbolic groups [8,38]; relatively hyperbolic groups [12]; Baumslag-Solitar groups BS(m, n) except for BS(1, 1) [13], generalized Baumslag-Solitar groups, that is, finitely generated groups which act on a tree with all edge and vertex stabilizers infinite cyclic [37]; the solvable generalization Γ of BS (1, n) given by the short exact sequence 1 → Z[ 1 n ] → Γ → Z k → 1, as well as any group quasi-isometric to Γ [52]; a wide class of saturated weakly branch groups (including the Grigorchuk group [27] and the Gupta-Sidki group [29]) [11], Thompson's groups F [2] and T [3,21]; generalized Thompson's groups F n, 0 and their finite direct products [23]; Houghton's groups [22,34]; symplectic groups Sp(2n, Z), the mapping class groups Mod S of a compact oriented surface S with genus g and p boundary components, 3g + p − 4 > 0, and the full braid groups B n (S) on n > 3 strands of a compact surface S in the cases where S is either the compact disk D, or the sphere S 2 [14]; some classes of Artin groups of infinite type [35]; extensions of SL(n, Z), PSL(n, Z), GL(n, Z), PGL(n, Z), Sp(2n, Z), PSp(2n, Z), n > 1, by a countable abelian group, and normal subgroups of SL(n, Z), n > 2, not contained in the center [40]; GL(n, K) and SL(n, K) if n > 2 and K is an infinite integral domain with trivial group of automorphisms, or K is an integral domain, which has a zero characteristic and for which Aut(K) is periodic [42]; irreducible lattices in a connected semisimple Lie group G with finite center and real rank at least 2 [41]; non-amenable, finitely generated residually finite groups [17] (this class gives a lot of new examples of groups with the R ∞ -property); some metabelian groups of the form Q n ⋊ Z and Z[1/p] n ⋊ Z [15]; lamplighter groups Z n ≀ Z if and only if 2|n or 3|n [25]; free nilpotent groups N rc of rank r = 2 and class c 9 [26], N rc of rank r = 2 or r = 3 and cl...…”

Sigma theory and twisted conjugacy, II: Houghton groups and pure symmetric automorphism groups

“…The above theorem was first proved by Burillo, Matucci, and Ventura [8]. Shortly thereafter, Gonçalves and Sankaran [15] also independently obtained the same result.…”

“…The main idea is to make use of homeomorphisms in F ix(ρ), whose supports have arbitrarily large number of disjoint intervals in S 1 . (This was also the idea used in the proof by Burillo, Matucci, and Ventura [8].) Proof.…”

“…The fact that T has the R ∞ property has been proved first by Burillo, Matucci, and Ventura [8] (see also [15]). The crucial point in the proofs of the result above is the same in both [8] and [15] and both the proofs rely on the description of the outer automorphism of T (recalled in Theorem 4.2 below).…”

“…The crucial point in the proofs of the result above is the same in both [8] and [15] and both the proofs rely on the description of the outer automorphism of T (recalled in Theorem 4.2 below). However, since the approaches before getting to the main point are slightly different, we provide our proof here which may contain some features that are useful for other situations (such as in Remark 4.7 below).…”