We give a characterization of the Anosov condition for reducible representations in terms of the eigenvalue magnitudes of the irreducible block factors of its block diagonalization. As in previous work, these Anosov representations comprise a collection of bounded convex domains in a finite-dimensional vector space, and this perspective allows us to conclude for many non-elementary hyperbolic groups that connected components of the character variety which consist entirely of Anosov representations do not contain reducible representations.

We study through the lens of Anosov representations the dynamical properties of reducible suspensions of linear representations of non-elementary hyperbolic groups, which are linear representations preserving and acting weakly unipotently on a proper non-zero subspace. We characterize when reducible suspensions are discrete and (almost) faithful, quasi-isometrically embedded, and Anosov. Anosov reducible suspensions correspond to points in bounded convex domains in a finite-dimensional real vector space. Stronger characterizations of such domains for symmetric Anosov representations allow us to find deformations of Borel Anosov representations which retain some but not all of the Anosov conditions and to compute examples of non-Anosov limits of Anosov representations.

In this note we prove an effective characterization of when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations, weakening the hypotheses to consider curves with explicitly bounded self-intersection number. As an application we show that for sufficiently large N, the set of unmarked traces associated to simple closed curves in a generically chosen representation to SL(N, R) distinguishes between pairs of non-isomorphic covers.

We prove that every closed, connected, orientable surface S of negative Euler characteristic admits a pair of finite-degree covers which are length isospectral over S but generically not simple length isospectral over S. To do this, we first characterize when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations. We also construct hyperbolic surfaces X and Y with the same full unmarked length spectrum but so that for each k, the sets of lengths associated to curves with at most k self-intersections differ.

Given two finite covers p: X -> S and q: Y -> S of a connected, oriented, closed surface S of genus at least 2, we attempt to characterize the equivalence of p and q in terms of which curves lift to simple curves. Using Teichmüller theory and the complex of curves, we show that two regular covers p and q are equivalent if for any closed curve γ on S, γ lifts a simple closed curve on X if and only if it does to Y. When the covers are abelian, we also give a characterization of equivalence in terms of which powers of simple closed curves lift to closed curves.

A collection Δ of simple closed curves on an orientable surface is an algebraic k-system if the algebraic intersection number ⟨ α, β ⟩ is k in absolute value for every α and β in Δ. Generalizing a theorem of [MRT14], we compute that the maximum size of an algebraic k-system of curves on a surface of genus g is 2g + 1 when g ≥ 3 or k is odd, and 2g otherwise. To illustrate the tightness in our assumptions, we present a construction of curves pairwise geometrically intersecting twice whose size grows as g2.