Commensurability and the character variety

On character varieties of two-bridge knot groups

“…Equation ( 15) also follows from the last equation of Lemma 3.4 and the relation tf n = f n+1 +f n−1 . Equation ( 17) also follows by subtracting 2n times the equation ( 15) from (16).…”

“…Recent results have shown that arithmetic properties of the SL 2 (C)-and PSL 2 (C)character varieties can give information about topological invariants such as the commensurability classes of knot complements ( [4], [15], [16]). Our irreducibility results allow us to use a criterion of Calegari and Dunfield [4] to prove a conjecture of Hoste and Shanahan [15,Conj.…”

Farey recursion and the character varieties for 2-bridge knots

“…We refer readers to [7] for background on SL 2 C character varieties and to [4], [10], or [21] for background on PSL 2 C character varieties.…”

Hasse-Weil zeta functions of ${\rm SL}_2$-character varieties of closed orientable hyperbolic $3$-manifolds

“…For a closed orientable hyperbolic 3-manifold of finite volume, the second cohomology group H 2 (π 1 (M), C 2 ) is non-zero in general since the abelianization π 1 (M) ab might have non-cyclic 2-torsion, namely there might be a representation ρ : π 1 (M) → PSL 2 (C) which never lifts to SL 2 (C) ([12], Lemma 2.3, see also [8]). Hence there might be a little difference in our case between the constructions of the PSL 2 (C)-character varieties in the references González-Acuña-Montesinos-Amilibia [12], Boyer-Zhang [4], Long-Reid [17] and Heusener-Porti [14].…”