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The extrinsic geometry of subgroups and the generalized word problem. (English) Zbl 0816.20032

The author introduces some functions on a finitely generated group \(G\) with a finitely generated subgroup \(H\) called the distortion, generalized isoperimetric and generalized isodiametric functions. The distortion function measures the deviation of the length of a word \(w\) representing an element of \(H\) in the alphabet \(X\) for \(G = \langle X \mid R\rangle\) from that in the alphabet \(Y\) for \(H\). If \[ w = y_{i_ 1} \ldots y_{i_ k} \prod_{j=1}^ N z_ j R_ j z_ j^{-1},\;\text{where }y_{j_ r} \in Y,\;z_ j \in F(X),\;R_ j \in R \cup R^{-1}, \] then \(g,h: \mathbb{N} \to \mathbb{N}\) are generalized isoperimetric and generalized isodiametric, respectively if for every \(w \in F(X)\) representing an element in \(H\) we have \(N \leq g(l(w))\) and \(l(z_ j) \leq h(l(w))\) for all \(j\). In a group \(G\) with solvable word problem the following conditions on a finitely generated subgroup are equivalent: 1. The generalized word problem (= the occurrence problem) for \(H\) in \(G\) is solvable. 2. The GIP (= generalized isoperimetric) function is recursive. 3. The GID function is recursive. 4. Some distortion function (for \(H\) in \(G\)) is recursive.
The author studies these functions and gives some examples quoting mainly the works of Mikhailova, Gersten and Gromov. There are two theorems with a number of corollaries. In the first of them the author gives certain conditions for two functions to be the distortion and the GIP functions for a subgroup \(H\) of a group \(G\) provided the quasiconvexity function of \(H < G\) is bounded by \({1\over 2}n - 1\). This latter condition is shown to be necessary and sufficient for a finitely generated subgroup of a group with recursive word problem to have solvable generalized word problem (we must also have the recursivity of a length). In another theorem a relationship is established between the functions of \(G\), \(H\) and \(G/H\) in the case where \(H\) is a normal subgroup of \(G\). There are also several open problems at the end of the paper.

MSC:

20F65 Geometric group theory
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20E07 Subgroup theorems; subgroup growth
20F05 Generators, relations, and presentations of groups
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