IA Scholar Query: Higman's Embedding Theorem in a General Setting and Its Application to Existentially Closed Algebras.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 30 Jun 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Computable analysis on the space of marked groups
https://scholar.archive.org/work/crcy2a5n55bp3hjydnr3lc5k5m
We investigate decision problems for groups described by word problem algorithms. This is equivalent to studying groups described by labelled Cayley graphs. We show that this corresponds to the study of computable analysis on the space of marked groups, and point out several results of computable analysis that can be directly applied to obtain group theoretical results. Those results, used in conjunction with the version of Higman's Embedding Theorem that preserves solvability of the word problem, provide powerful tools to build finitely presented groups with solvable word problem but with various undecidable properties. We also investigate the first levels of an effective Borel hierarchy on the space of marked groups, and show that on many group properties usually considered, this effective hierarchy corresponds sharply to the Borel hierarchy. Finally, we prove that the space of marked groups is a Polish space that is not effectively Polish. Because of this, many of the most important results of computable analysis cannot be applied to the space of marked groups. This includes the Kreisel-Lacombe-Schoenfield-Ceitin Theorem and a theorem of Moschovakis. The space of marked groups constitutes the first natural example of a Polish space that is not effectively Polish.Emmanuel Rauzywork_crcy2a5n55bp3hjydnr3lc5k5mThu, 30 Jun 2022 00:00:00 GMTUnsolved Problems in Group Theory. The Kourovka Notebook
https://scholar.archive.org/work/fhii5oyzvrb7vpeun6rsfapu2i
This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965. This is the 20th edition, which contains 126 new problems and a number of comments on problems from the previous editions.E. I. Khukhro, V. D. Mazurovwork_fhii5oyzvrb7vpeun6rsfapu2iMon, 27 Jun 2022 00:00:00 GMTOn the Boolean dimension of a graph and other related parameters
https://scholar.archive.org/work/tn5ydqkx4fgnhik2gwllsstl2a
We present the Boolean dimension of a graph, we relate it with the notions of inner, geometric and symplectic dimensions, and with the rank and minrank of a graph. We obtain an exact formula for the Boolean dimension of a tree in terms of a certain star decomposition. We relate the Boolean dimension with the inversion index of a tournament.Maurice Pouzet, Hamza Si Kaddour, Bhalchandra D. Thattework_tn5ydqkx4fgnhik2gwllsstl2aThu, 31 Mar 2022 00:00:00 GMTComputability and l2-Betti Numbers
https://scholar.archive.org/work/7glorpnsy5akzlots2xscy6mie
In Chapter 1, we will introduce L2 -Betti numbers after covering the preliminaries for this definition. We will also show an algebraic characterisation of L2 -Betti numbers: For a group G, all L2 -Betti numbers arising from G are given as dimRG ker(·A) for some self-adjoint A ∈ Mn×n(ZG) (see Section 1.2.4). In Section 1.3, we will cover Atiyah's conjecture and Lück's approximation theorem. Chapter 2 is dedicated to the introduction of computability concepts. After a 'naive' introduction into this subject, we will define different computability classes such as EC (effectively computable), LC (left-computable) and RC (right-computable). We will then take a look at some results on right-computability of topological invariants (Section 2.3). The main part of this thesis is Chapter 3. We will start with a survey on some known computability results on L2 -Betti numbers (Section 3.1). We will then discuss right-, left- and effective computability of L2 -Betti numbers under different assumptions. Finally, in Chapter 4, we will discuss an implementation of some of the main results in the Lean Theorem Prover. This formally verifies some of these results. The .lean files used for this can be found on a git repository online. More information on how to install these files can be found in Section 4.2.Matthias Uscholdwork_7glorpnsy5akzlots2xscy6mie