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On subgroup distortion in finitely presented groups. (English. Russian original) Zbl 0905.20020
Sb. Math. 188, No. 11, 1617-1664 (1997); translation from Mat. Sb. 188, No. 1, 51-98 (1997).
Let $$G$$ be a group generated by a finite set $${\mathcal A}=\{a_1,\dots,a_m\}$$. Then the length $$| g|=| g|_{\mathcal A}$$ of an element $$g$$ is the length $$\| w\|$$ of the shortest word $$w$$ on the alphabet $${\mathcal A}^{\pm 1}$$ representing the element $$g$$. If there exists an isomorphic embedding $$G\hookrightarrow H$$ and $$H$$ is generated by a set $${\mathcal B}=\{b_1,\dots, b_n\}$$ then $$| g|_{\mathcal B}=c| g|_{\mathcal A}$$ for some constant $$c$$ independent of the element $$g$$. In particular, if $$G=\langle{\mathcal A}\rangle=\langle{\mathcal B}\rangle$$, then there exist positive constants $$c_1,c_2$$ such that $$c_1| g|_{\mathcal A}\leq| g|_{\mathcal B}\leq c_2| g|_{\mathcal A}$$ for every $$g\in G$$.
The functions $$\ell_1,\ell_2\colon G\to\mathbb{N}$$ are said to be equivalent if there are positive constants $$c_1,c_2$$ such that $$c_1\ell_1(g)\leq\ell_2(g)\leq c_2\ell_1(g)$$ for every element $$g\in G$$. A function $$\ell\colon G\to\mathbb{N}$$ is said to be computable if for each representation of an arbitrary element $$g$$ as a word on some generating set one can algorithmically compute $$\ell(g)$$.
Theorem 2. Let $$\ell\colon G\to\mathbb{N}$$ be a computable function satisfying the following conditions: (D1) $$\ell(g)=\ell(g^{-1})$$ for every $$g\in G$$ and $$\ell(g)=0$$ if and only if $$g=1$$; (D2) $$\ell(gh)=\ell(g)+\ell(h)$$, $$g,h\in G$$; (D3) there exists a positive number $$a$$ such that $$\text{card}\{g\in G\mid\ell(g)\leq r\}\leq a^r$$ for every $$r\in\mathbb{N}$$. Then there exists an isomorphic embedding $$G\hookrightarrow H$$ where $$H$$ is a finitely presented group such that $$\ell$$ is equivalent to the restriction of $$|\;|_H$$ to $$G$$.
The following theorem is a refinement of Higman’s theorem. Theorem 3. Let $$R$$ be a group with a finite set of generators $$\mathcal A$$ and a recursively enumerable set of (defining) relations. Then there exists an isomorphic embedding $$R\hookrightarrow H$$ where $$H$$ has a finite set of generators $$\mathcal B$$ and finite set of defining relations such that $$| g|_{\mathcal A}=| g|_{\mathcal B}$$ for any $$g\in G$$.
The following theorem gives a possibility to choose a universal group $$U$$. Theorem 4. There exists a group $$U$$ having a finite set of generators $$\mathcal B$$ and finite set of defining relations satisfying the following property: for every group $$H$$ with finite set of generators $$\mathcal A$$ and a recursively enumerable set of defining relations there exists an isomorphic embedding $$H\hookrightarrow U$$ such that $$| g|_{\mathcal A}\leq| g|_{\mathcal B}$$ for all $$g\in H$$.

##### MSC:
 20F05 Generators, relations, and presentations of groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20F65 Geometric group theory 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20E07 Subgroup theorems; subgroup growth 03D20 Recursive functions and relations, subrecursive hierarchies
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