Group Embeddings with Algorithmic Properties

Abstract

We show that every countable group H with solvable word problem can be subnormally embedded into a 2-generated group G which also has solvable word problem. Moreover, the membership problem for H < G is also solvable. We also give estimates of time and space complexity of the word problem in G and of the membership problem for H < G. IntroductionIn the famous paper [8] by Higman, B.H.Neumann and H. Neumann in 1949, using constructions based on HNN-extensions, it was shown that every countable group can be embedded in a group generated by two elements. Later, B.H. Neumann and H. Neumann suggested an alternative embedding construction based on wreath products [15], which allowed them to show that every countable solvable group can be embedded in a 2-generated solvable group. Further development of these ideas was done by Hall [7]. Subsequently, other constructions based on this ideas were introduced, where the 2-generated group inherited some other properties of the initial group. Most of the aforementioned embedding constructions were motivated either by the desire to better control the algebraic structure or geometric properties of the embedding, but none of the constructions based on wreath products was

Karol Borsuk 1905–1982

“…This is a part of the statement of Theorem 1 of [10] and the definition of φ 1 at line 19 on p.4927 of that paper. We only add that applying this theorem • we consider our G with respect to the generators given by any enumeration of G = {a (1) , a (2) , .…”

Computability, Orders, and Solvable Groups

“…The main technical tool of our paper is the following embedding theorem, which, in a weaker form and using a slightly different language, can be found in [2], and is based on a modified version of B. Neumann's construction from [9]. We will prove this theorem in Section 4.…”

“…Proof of Theorem 3. The proof of Theorem 3 uses the embedding construction described in [2]. Moreover, the proof is in a way a commentary on the proof of the main theorem from [2].…”

“…The proof of Theorem 3 uses the embedding construction described in [2]. Moreover, the proof is in a way a commentary on the proof of the main theorem from [2]. There are also differences in notations compared with the exposition from [2].…”

“…Moreover, the proof is in a way a commentary on the proof of the main theorem from [2]. There are also differences in notations compared with the exposition from [2]. For example, the embedding Φ X described below is the embedding φ from [2], the map f defined below is c in [2], f i is b (i) , etc.…”