Twisted conjugacy classes in symplectic groups, mapping class groups and braid groups (with an appendix written jointly with Francois Dahmani)
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Citations
The R ∞ and S ∞ properties for linear algebraic groups
Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups
Twisted conjugacy classes in general and special linear groups
Twisted conjugacy classes in nilpotent groups
Twisted conjugacy classes in lattices in semisimple lie groups
References
Reidemeister number of any automorphism of a Gromov hyperbolic group is infinite
Twisted conjugacy and quasi‐isometry invariance for generalized solvable Baumslag–Solitar groups
Twisted Burnside theorem for type II_1 groups: an example
Twisted conjugacy classes in R. Thompson's group F
Twisted conjugacy and quasi-isometry invariance for generalized solvable Baumslag-Solitar groups
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Frequently Asked Questions (8)
Q2. What is the R property for Bn(S 2)?
Since the center of Bn(S 2) is a characteristic subgroup, φ induces a homomorphism of the short exact sequence1 → Z2 → Bn(S 2) →ModS2n → 1where the short exact sequence was obtained from the sequence in Theorem 2.3.
Q3. What is the symplectic group of order 2n?
Let φ be the automorphism which is the conjugation by the diagonal matrix of order 2n, where the elements of the diagonal are ai,i = (−1) i+1.
Q4. What is the definition of the group Sp(2n,Z)?
Recall that the elements of the the group Sp(2n,Z) are automorphisms which are obtained as the induced homomorphisms in H1(S,Z) by an orientation preserving homeomorphisms of the orientable closed surface S of genus n.The authors refer to [42] and [44] for most of the properties of the group of symplectic matrices.
Q5. What is the r property of the mapping class group of closed surface?
For S = S 2 the authors have ModS = {1}, the trivial group, and Mod ∗ S = Z2, therefore Out(Mod ∗ S) = Out(ModS) = 1.The authors will show that the mapping class group of closed surface has the R∞property.
Q6. what is the matrices of order 2n?
The product Mw̄(φ(M)−1 is of the form( A 00 I2n−2)where the A is of order 2 × 2, I2n−2 is the identity matrix of order 2n − 2, and 0′s are the trivial matrices of orders 2× 2n− 2, 2n− 2× 2, respectively.
Q7. what is the v of the column?
If v is any of the above column, then the inner product of (a1,1, a1,2, a1,3, a1,4, ....., a1,2n−1, a1,2n) with the column vector (0, 0, v) is zero.
Q8. What is the origin of the interest in twisted conjugacy relations?
The interest in twisted conjugacy relations has its origins, in particular, in the NielsenReidemeister fixed point theory (see, e.g. [43, 35, 12]), in Selberg theory (see, eg. [46, 1]), and in Algebraic Geometry (see, e.g. [30]).