We give sufficient conditions for a first order expansion of the real line to define the standard model of the monadic second order theory of one successor. Such an expansion does not satisfy any of the combinatorial tameness properties defined by Shelah, such as NIP or even NTP2. We use this to deduce the first general results about definable sets in NTP2 expansions of (R, <,+). The goal of th...

Marden conjectured that a hyperbolic 3-manifold M with finitely generated fundamental group is tame, i.e. it is homeomorphic to the interior of a compact manifold with boundary [42]. Since then, many consequences of this conjecture have been developed by Kleinian group theorists and 3-manifold topologists. We prove this conjecture in theorem 10.2, actually in slightly more generality for PNC ma...

We show that Shelah’s Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ tame and applying the categoricity transfer of Grossberg and VanDieren [GV06a]. These techniques also apply t...

A Fréchet space, X , with a sequence of generating seminorms {‖·‖k} ∞ k=1 is called tame in case there exists an increasing function σ : N → N, such that for every continuous linear operator T from X into itself, there exists an N0 and C > 0 such that ‖T (x)‖n ≤ C‖x‖σ(n), ∀x ∈ X and n ≥ N0. This property does not depend upon the choice of fundamental system of semi-norms for X and is a property...

A 1-combing for a finitely presented group consists of a continuous family of paths based at the identity and ending at points x in the 1-skeleton of the Cayley 2-complex associated to the presentation. We define two functions (radial and ball tameness functions) that measure how efficiently a 1-combing moves away from the identity. These functions are geometric in the sense that they are quasi...

We show that various tameness assertions about abstract elementary classes imply the existence of large cardinals under mild cardinal arithmetic assumptions. For instance, we show: Theorem. Let κ be uncountable such that μω < κ for every μ < κ. If every AEC with Löwenheim-Skolem number less than κ is < κ-tame, then κ is almost strongly compact. This is done by isolating a class of AECs that exh...

Antipredatory defenses are maintained when benefit exceeds cost. A weak predation pressure may lead insular lizards to tameness. Podarcis lilfordi exhibits a high degree of insular tameness, which may explain its extinction from the main island of Menorca when humans introduced predators. There are three species of lizards in Menorca: the native P. lilfordi, only on the surrounding islets, and ...

Domestication has caused a range of similar phenotypic changes across taxa, relating to physiology, morphology and behaviour. It has been suggested that this recurring domesticated phenotype may be a result of correlated responses to a central trait, namely increased tameness. We selected Red Junglefowl, the ancestors of domesticated chickens, during five generations for reduced fear of humans....