Abstract An equicritical stratum is the locus of univariate monic squarefree complex polynomials where the critical points have prescribed multiplicities. Tracking the positions of both roots and critical points, there is a natural “monodromy map” taking the fundamental group into a braid group. We show here that when there are exactly two critical points, this monodromy map is noninjective.

Abstract In this paper we provide a means of certifying infinitesimal projective rigidity relative to the cusp for hyperbolic once-punctured torus bundles in terms of the Wada Invariant of representations associated to the holonomy. We also relate this polynomial to an induced action on the tangent space of the character variety of the free group of rank 2 into $$\text {PGL}(4,\mathbb {R})$$ PGL ( 4 , R ) that arises from the holonomy of a hyperbolic once-punctured torus bundle. We prove the induced action on the tangent space of the character variety is the same as the group theoretic action that arises from the monodromy action of the once-punctured torus bundle on cohomology.

Abstract For a surface S of sufficient complexity, Dehn twists act elliptically on the arc, curve, or relative arc graph of S. We show that composing a Dehn twist with a shift map results in a loxodromic isometry of the relative arc graph $$\mathcal {A}(S,p)$$ A ( S , p ) for any surface S with an isolated puncture p admitting a shift map. Therefore, shift maps are not type-preserving.

Abstract Using structural properties of groups of small order, we establish the non-existence of varieties isogenous to a higher product of dimension n greater than 3 with fixed topological Euler number $$(-2)^n$$ ( - 2 ) n and trivial first Betti number.