Abstract We show that the relative cohomological dimension $$\textrm{cd}_R(G,H)$$ cd R ( G , H ) of a relatively hyperbolic pair ( G , H ) is always finite when G does not contain R -torsion. We also show that this dimension is preserved under quasi-isometries, provided that G is torsion-free and the peripheral subgroup H is unconstricted and of type $$F_{\infty }$$ F ∞ . As a corollary of our methods, we compute $$\textrm{cd}_{\mathbb {Z}}(G,H)$$ cd Z ( G , H ) in several cases.

Abstract We extend Wood’s graph theoretic interpretation of certain quotients of the mod 2 dual Steenrod algebra to quotients of the mod p dual Steenrod algebra where p is an odd prime and to quotients of the $$C_2$$ C 2 -equivariant dual Steenrod algebra. We establish connectedness criteria for graphs associated to monomials in these algebra quotients and investigate questions about trees and Hamilton cycles in these settings. We also give graph theoretic interpretations of algebraic structures such as the coproduct and antipode arising from the Hopf algebra structure on the mod p dual Steenrod algebra and the Hopf algebroid structure of the $$C_2$$ C 2 -equivariant dual Steenrod algebra.

Abstract We show that for a large class of $$\infty $$ ∞ -topoi there exist unstable arithmetic fracture squares, i.e. squares which recover a nilpotent sheaf F as the pullback of the rationalization of F with the product of the p -completions of F ranging over all primes $$p\in {\mathbb {Z}}$$ p ∈ Z .

Abstract The Morava E -theories, $$E_{n}$$ E n , are complex-oriented 2-periodic ring spectra, with homotopy groups $$W_{{{\mathbb {F}}}_{p^{n}}}[[u_{1}, u_{2},\ldots , u_{n-1}]][u,u^{-1}]$$ W F p n [ [ u 1 , u 2 , … , u n - 1 ] ] [ u , u - 1 ] . Here W denotes the ring of Witt vectors. $$E_{n}$$ E n is a Landweber exact spectrum and hence uniquely determined by its homotopy groups as $$BP_{*}$$ B P ∗ -algebra. Algebraic K -theory of $$E_{n}$$ E n is a key ingredient towards analyzing the layers in the p -complete Waldhausen’s algebraic K -theory chromatic tower. One hopes to use the machinery of trace methods to get results towards algebraic K -theory once the computation for $$THH(E_{n})$$ T H H ( E n ) is known. In this paper we describe $$THH(E_{2})$$ T H H ( E 2 ) as part of consecutive chain of cofiber sequences where each cofiber sits in the next cofiber sequence and the first term of each cofiber sequence is describable completely in terms of suspensions and localizations of $$E_{2}$$ E 2 . For these results, we first calculate K ( i )-homology of $$THH(E_{2})$$ T H H ( E 2 ) using a Bökstedt spectral sequence and then lift the generating classes of K (1)-homology to fundamental classes in homotopy group of $$THH(E_{2})$$ T H H ( E 2 ) . These lifts allow us to construct terms of the cofiber sequence and explicitly understand how they map to $$THH(E_{2})$$ T H H ( E 2 ) .

Abstract We extend a CDGA V with a perfect pairing of degree n on cohomology to a CDGA $$\hat{V}$$ V ^ with a pairing of degree n on chain level such that $$\hat{V}$$ V ^ admits a Hodge decomposition and retracts onto V preserving the pairing on cohomology; here we suppose that V is either 1-connected, or that V is connected, of finite type, and n is odd. We show that a Hodge decomposition of $$\hat{V}$$ V ^ induces a differential Poincaré duality model of V in a natural way. Assuming that $$\textrm{H}(V)$$ H ( V ) is 1-connected, we apply our extension to a Sullivan model of V in the proof of the existence and “uniqueness” of a 1-connected differential Poincaré duality model of V by Lambrechts & Stanley; we eliminate their extra assumptions in the uniqueness statement, including $$\textrm{H}^2(V)=0$$ H 2 ( V ) = 0 if n is odd.

Abstract In this note I give a conceptual proof of the fact that the mod 2 dual Steenrod algebra corepresents the group scheme of strict automorphisms of the formal additive group over $${\mathbb {F}}_2$$ F 2 . Contrary to existing proofs, it does not use the $$E_\infty $$ E ∞ -structure of $$H{\mathbb {F}}_2$$ H F 2 (Steenrod operations), nor does it proceed by producing a generators-and-relations presentation by some explicit calculation. Instead it relies on universal properties of bordism spectra, thus giving a stronger conceptual foundation for what is arguably the first instance of the well-studied deep connection between the algebraic geometry of formal groups and the stable homotopy category.