Twisted conjugacy in simply connected groups

If G is a group and an automorphism of G, one has the twisted conjugation action of G on itself This paper collects a number of results — more or less well known — for the case that G is a simply connected semisimple group.     

Twisted conjugacy classes in nilpotent groups

Let N be a finitely generated nilpotent group. We show that there is an algorithm that for any automorphism φ∈Aut(N) computes its Reidemeister number R(φ). It is proved that any free nilpotent group N_rc of rank r and class c belongs to class R_∞ if any of the following conditions holds: r=2 and c≥4; r=3 and c≥12; r≥4 and c≥2r.     

Twisted conjugacy classes in symplectic groups,mapping class groups and braid groups (with an appendix written jointly with Francois Dahmani)

We prove that the symplectic group Sp(2n,\mathbbZ){Sp(2n,\mathbb{Z})} and the mapping class group Mod _S of a compact surface S satisfy the R _∞ property. We also show that B _n (S), the full braid group on n-strings of a surface S, satisfies the R _∞ property in the cases where S is either the compact disk D, or the sphere S ². This means that for any automorphism f{\phi} of G, where G is one of the above groups, the number of twisted f{\phi}-conjugacy classes is infinite.     

Twisted conjugacy classes in saturated weakly branch groups

For a wide class of saturated weakly branch groups, including the (first) Grigorchuk group and the Gupta-Sidki group, we prove that the Reidemeister number of any automorphism is infinite.      

On braid groups

A dissertation in the Department of Mathematics submitted to the Faculty of the Graduate School of Arts and Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy at New York University.     

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).     

On braid groups and right-angled Artin groups

In this article we prove a special case of a conjecture of A. Abrams and R. Ghrist about fundamental groups of certain aspherical spaces. Specifically, we show that the \(n\) -point braid group of a linear tree is a right-angled Artin group for each \(n\) .     

Parabolic conjugacy in general linear groups

Let q be a power of a prime and n a positive integer. Let P(q) be a parabolic subgroup of the finite general linear group GL _n (q). We show that the number of P(q)-conjugacy classes in GL _n (q) is, as a function of q, a polynomial in q with integer coefficients. This answers a question of Alperin in (Commun. Algebra 34(3): 889–891, 2006)     

Local representations of braid groups

We study the local linear representations of the braid group B ₃ and the homogeneous local representations of B _n for n ≥ 2. We investigate the connection of these representations with the Burau representation. The linear representations of B _n are constructed from the Wada representation of B _n in the automorphism group of a free group.     

On conjugacy classes in metaplectic groups

Let E be a finite dimensional symplectic space over a local field of characteristic zero. We show that for every element in the metaplectic double cover of the symplectic group Sp(E), and are conjugate by an element of GSp(E) with similitude −1.     

Embedding right-angled Artin groups into graph braid groups

We construct an embedding of any right-angled Artin group G(Δ) defined by a graph Δ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of Δ. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.      

Bianchi groups are conjugacy separable

We prove that non-uniform arithmetic lattices of SL₂(C) and consequently the Bianchi groups are conjugacy separable. The proof is based on recent deep results of Agol, Long, Reid and Minasyan. The conjugacy separability of groups commensurable with limit groups is also established.     

On the cycling operation in braid groups

The cycling operation is a special kind of conjugation that can be applied to elements in Artin’s braid groups, in order to reduce their length. It is a key ingredient of the usual solutions to the conjugacy problem in braid groups. In their seminal paper on braid-cryptography, Ko, Lee et al. proposed the cycling problem as a hard problem in braid groups that could be interesting for cryptography. In this paper we give a polynomial solution to that problem, mainly by showing that cycling is surjective, and using a result by Maffre which shows that pre-images under cycling can be computed fast. This result also holds in every Artin-Tits group of spherical type, endowed with the Artin Garside structure.On the other hand, the conjugacy search problem in braid groups is usually solved by computing some finite sets called (left) ultra summit sets (left-USSs), using left normal forms of braids. But one can equally use right normal forms and compute right-USSs. Hard instances of the conjugacy search problem correspond to elements having big (left and right) USSs. One may think that even if some element has a big left-USS, it could possibly have a small right-USS. We show that this is not the case in the important particular case of rigid braids. More precisely, we show that the left-USS and the right-USS of a given rigid braid determine isomorphic graphs, with the arrows reversed, the isomorphism being defined using iterated cycling. We conjecture that the same is true for every element, not necessarily rigid, in braid groups and Artin-Tits groups of spherical type.     

Affine weyl groups and conjugacy classes in Weyl groups

We define a map from an affine Weyl group to the set of conjugacy classes of an ordinary Weyl group. Supported in part by the National Science Foundation.