Commensurability classes of fake quadrics

Commensurability Classes of Twist Knots

In this paper we prove that if MK is the complement of a non-fibered twist knot K in S, then MK is not commensurable to a fibered knot complement in a Z/2Z-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In ad...

Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds

This sharpens [10], where it was shown that the complex length spectrum of M determines its commensurability class. Suppose M ′ is an arithmetic hyperbolic 3-manifold which is not commensurable to M . Theorem 1.1 implies QL(M) 6= QL(M ′), though by Example 2.1 below it is possible that one of QL(M ′) or QL(M) contains the other. By the length formulas recalled in §2.1 and §2.2, each element of ...

COMMENSURABILITY CLASSES OF (−2, 3, n) PRETZEL KNOT COMPLEMENTS

Let K be a hyperbolic (−2, 3, n) pretzel knot and M = S \ K its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of M . Indeed, if n 6= 7, we show that M is the unique knot complement in its class. We include examples to illustrate how our methods apply to a broad class of Montesinos knots.

Fake quadrics from irreducible lattices acting on the product of upper half planes

In the present article, we provide examples of fake quadrics, i.e. complex surfaces of general type with the same numerical invariants as the quadric P × P which are quotients of the bidisc by an irreducible lattice of automorphisms. Moreover we list classes of arithmetic lattices over a real quadratic number field which define a fake quadric and give general results towards a classification of...

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