Special transformations in algebraically closed valued fields

We present two of the three major steps in the construction of motivic integration, that is, a homomorphism between Grothendieck semigroups that are associated with a first-order theory of algebraically closed valued fields, in the fundamental work of Hrushovski and Kazhdan. We limit our attention to a simple major subclass of V-minimal theories of the form ACVF_S(0, 0), that is, the theory of algebraically closed valued fields of pure characteristic $0$ expanded by a (VF, Gamma)-generated substructure S in the language L_RV. The main advantage of this subclass is the presence of syntax. It enables us to simplify the arguments with many different technical details while following the major steps of the Hrushovski-Kazhdan theory.Comment: This is the published version of a part of the notes on the Hrushovski-Kazhdan integration theory. To appear in the Annals of Pure and Applied Logi

Maximum Orders of Cyclic and Abelian Extendable Actions on Surfaces

Let $\Sigma_g (g>1)$ be a closed surface embedded in $S^3$. If a group $G$ can acts on the pair $(S^3, \Sigma_g)$, then we call such a group action on $\Sigma_g$ extendable over $S^3$. In this paper we show that the maximum order of extendable cyclic group actions is $4g+4$ when $g$ is even and $4g-4$ when $g$ is odd; the maximum order of extendable abelian group actions is $4g+4$. We also give results of similar questions about extendable group actions over handlebodies.Comment: 22pages, 10 figure

Alternating Heegaard diagrams and Williams solenoid attractors in 3--manifolds

We find all Heegaard diagrams with the property "alternating" or "weakly alternating" on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two 3--manifolds, each admits an automorphism whose non-wondering set consists of two Williams solenoids, one attractor and one repeller. These manifolds contain half of Prism manifolds, Poincar\'e's homology 3--sphere and many other Seifert manifolds, all integer Dehn surgeries on the figure eight knot, also many connected sums. The result shows that many kinds of 3--manifolds admit a kind of "translation" with certain stability.Comment: 26 pages, 44 figure

Quantifier elimination for the reals with a predicate for the powers of two

In 1985, van den Dries showed that the theory of the reals with a predicate for the integer powers of two admits quantifier elimination in an expanded language, and is hence decidable. He gave a model-theoretic argument, which provides no apparent bounds on the complexity of a decision procedure. We provide a syntactic argument that yields a procedure that is primitive recursive, although not elementary. In particular, we show that it is possible to eliminate a single block of existential quantifiers in time $2^0_{O(n)}$, where $n$ is the length of the input formula and $2_k^x$ denotes $k$-fold iterated exponentiation