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| In the mathematical subject of [[group theory]], the '''Stallings theorem about ends of groups''' states that a [[finitely generated group]] ''G'' has more than one end if and only if the group ''G'' admits a nontrivial decomposition as an [[free product with amalgamation|amalgamated free product]] or an [[HNN extension]] over a finite [[subgroup]]. In the modern language of [[Bass–Serre theory]] the theorem says that a finitely generated group ''G'' has more than one end if and only if ''G'' admits a nontrivial (that is, without a global fixed point) [[group action|action]] on a simplicial [[tree (graph theory)|tree]] with finite edge-stabilizers and without edge-inversions.
| | Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. What I advise you do next is save the backup data file to a remote place like a CD-ROM, external disk drive if you have one or a provider such as Dropbox. One really cool features about this amazing and free wp plugin is that the code it generates is completely portable. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. Understanding how Word - Press works can be a challenge, but it is not too difficult when you learn more about it. <br><br>If you adored this article therefore you would like to get more info pertaining to [http://scridle.nl/wordpress_backup_424416 wordpress backup] i implore you to visit our web site. Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. Our Daily Deal Software plugin brings the simplicity of setting up a Word - Press blog to the daily deal space. So if you want to create blogs or have a website for your business or for personal reasons, you can take advantage of free Word - Press installation to get started. Word - Press makes it possible to successfully and manage your website. <br><br>You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. You can now search through the thousands of available plugins to add all kinds of functionality to your Word - Press site. User friendly features and flexibility that Word - Press has to offer is second to none. So, if you are looking online to hire dedicated Wordpress developers, India PHP Expert can give a hand you in each and every best possible way. <br><br>There has been a huge increase in the number of developers releasing free premium Word - Press themes over the years. Russell HR Consulting provides expert knowledge in the practical application of employment law as well as providing employment law training and HR support services. Enterprise, when they plan to hire Word - Press developer resources still PHP, My - SQL and watch with great expertise in codebase. It supports backup scheduling and allows you to either download the backup file or email it to you. Fortunately, Word - Press Customization Service is available these days, right from custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project. <br><br>Yet, overall, less than 1% of websites presently have mobile versions of their websites. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Just download it from the website and start using the same. It is a fact that Smartphone using online customers do not waste much of their time in struggling with drop down menus. Likewise, professional publishers with a multi author and editor setup often find that Word - Press lack basic user and role management capabilities. |
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| The theorem was proved by [[John R. Stallings]], first in the [[torsion-free group|torsion-free]] case (1968)<ref>John R. Stallings. [http://www.jstor.org/pss/1970577 ''On torsion-free groups with infinitely many ends.''] [[Annals of Mathematics]] (2), vol. 88 (1968), pp. 312–334</ref> and then in the general case (1971).<ref>John Stallings. ''Group theory and three-dimensional manifolds.'' A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. Yale Mathematical Monographs, 4. Yale University Press, New Haven, Conn.-London, 1971.</ref>
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| ==Ends of graphs==
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| {{main|End (graph theory)}}
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| Let Γ be a connected [[graph (mathematics)|graph]] where the degree of every vertex is finite. One can view Γ as a [[topological space]] by giving it the natural structure of a one-dimensional [[cell complex]]. Then the ends of Γ are the [[End (topology)|ends]] of this topological space. A more explicit definition of the number of [[End (graph theory)|ends of a graph]] is presented below for completeness.
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| Let ''n'' ≥ 0 be a non-negative integer. The graph Γ is said to satisfy ''e''(Γ) ≤ ''n'' if for every finite collection ''F'' of edges of Γ the graph Γ − ''F'' has at most ''n'' infinite [[Connected component (graph theory)|connected components]]. By definition, ''e''(Γ) = ''m'' if ''e''(Γ) ≤ ''m'' and if for every 0 ≤ ''n'' < ''m'' the statement ''e''(Γ) ≤ ''n'' is false. Thus ''e''(Γ) = ''m'' if ''m'' is the smallest nonnegative integer ''n'' such that ''e''(Γ) ≤ ''n''. If there does not exist an integer ''n'' ≥ 0 such that ''e''(Γ) ≤ ''n'', put ''e''(Γ) = ∞. The number ''e''(Γ) is called ''the number of ends of'' Γ.
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| Informally, ''e''(Γ) is the number of "connected components at infinity" of Γ. If ''e''(Γ) = ''m'' < ∞, then for any finite set ''F'' of edges of Γ there exists a finite set ''K'' of edges of Γ with ''F'' ⊆ ''K'' such that Γ − ''F'' has exactly ''m'' infinite connected components. If ''e''(Γ) = ∞, then for any finite set ''F'' of edges of Γ and for any integer ''n'' ≥ 0 there exists a finite set ''K'' of edges of Γ with ''F'' ⊆ ''K'' such that Γ − ''K'' has at least ''n'' infinite connected components.
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| ==Ends of groups==
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| Let ''G'' be a [[finitely generated group]]. Let ''S'' ⊆ ''G'' be a finite [[generating set of a group|generating set]] of ''G'' and let Γ(''G'', ''S'') be the [[Cayley graph]] of ''G'' with respect to ''S''. The ''number of ends of'' ''G'' is defined as ''e''(''G'') = e(Γ(''G'', ''S'')). A basic fact in the theory of ends of groups says that e(Γ(''G'', ''S'')) does not depend on the choice of a finite [[generating set of a group|generating set]] ''S'' of ''G'', so that ''e''(''G'') is well-defined.
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| ===Basic facts and examples===
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| *For any [[finitely generated group]] ''G'' we have ''e''(''G'') ∈ {0, 1, 2, ∞}
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| *For a [[finitely generated group]] ''G'' we have ''e''(''G'') = 0 if and only if ''G'' is finite.
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| *For the [[infinite cyclic group]] <math>\mathbb Z</math> we have <math>e(\mathbb Z)=2.</math>
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| *For the [[free abelian group]] of rank two <math>\mathbb Z^2</math> we have <math>e(\mathbb Z^2)=1.</math>
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| *For a [[free group]] ''F''(''X'') where 1 < |''X''| < ∞ we have ''e''(''F''(''X'')) = ∞
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| *For a [[finitely generated group]] ''G'' we have ''e''(''G'') = 2 if and only if ''G'' is [[virtually]] [[infinite cyclic group|infinite cyclic]] (that is, ''G'' contains an infinite cyclic [[subgroup]] of finite [[index of a subgroup|index]]).
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| ===Cuts and almost invariant sets===
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| Let ''G'' be a [[finitely generated group]], ''S'' ⊆ ''G'' be a finite [[generating set of a group|generating set]] of ''G'' and let Γ = Γ(''G'', ''S'') be the [[Cayley graph]] of ''G'' with respect to ''S''. For a subset ''A'' ⊆ ''G'' denote by ''A''<sup>∗</sup> the complement ''G'' − ''A'' of ''A'' in ''G''.
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| For a subset ''A'' ⊆ ''G'', the ''edge boundary'' or the ''co-boundary'' ''δA'' of ''A'' consists of all (topological) edges of Γ connecting a vertex from A with a vertex from ''A''<sup>∗</sup>.
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| Note that by definition ''δA'' = ''δA''<sup>∗</sup>.
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| An ordered pair (''A'', ''A''<sup>∗</sup>) is called a ''cut'' in Γ if ''δA'' is finite. A cut (''A'',''A''<sup>∗</sup>) is called ''essential'' if both the sets ''A'' and ''A''<sup>∗</sup> are infinite.
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| A subset ''A'' ⊆ ''G'' is called ''almost invariant'' if for every ''g''∈''G'' the [[symmetric difference]] between ''A'' and ''Ag'' is finite. It is easy to see that (''A'', ''A''<sup>∗</sup>) is a cut if and only if the sets ''A'' and ''A''<sup>∗</sup> are almost invariant (equivalently, if and only if the set ''A'' is almost invariant).
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| ====Cuts and ends====
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| A simple but important observation states:
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| :''e''(''G'') > 1 if and only if there exists at least one essential cut (''A'',''A''<sup>∗</sup>) in Γ.
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| ====Cuts and splittings over finite groups====
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| If ''G'' = ''H''∗''K'' where ''H'' and ''K'' are nontrivial [[finitely generated group]]s then the [[Cayley graph]] of ''G'' has at least one essential cut and hence ''e''(''G'') > 1. Indeed, let ''X'' and ''Y'' be finite generating sets for ''H'' and ''K'' accordingly so that ''S'' = ''X'' ∪ ''Y'' is a finite generating set for ''G'' and let Γ=Γ(''G'',''S'') be the [[Cayley graph]] of ''G'' with respect to ''S''. Let ''A'' consist of the trivial element and all the elements of ''G'' whose normal form expressions for ''G'' = ''H''∗''K'' starts with a nontrivial element of ''H''. Thus ''A''<sup>∗</sup> consists of all elements of ''G'' whose normal form expressions for ''G'' = ''H''∗''K'' starts with a nontrivial element of ''K''. It is not hard to see that (''A'',''A''<sup>∗</sup>) is an essential cut in Γ so that ''e''(''G'') > 1.
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| A more precise version of this argument shows that for a [[finitely generated group]] ''G'':
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| *If ''G'' = ''H''∗<sub>''C''</sub>''K'' is a [[free product with amalgamation]] where ''C'' is a finite group such that ''C'' ≠ ''H'' and ''C'' ≠ ''K'' then ''H'' and ''K'' are finitely generated and ''e''(''G'') > 1 .
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| *If <math>\scriptstyle G=\langle H, t| t^{-1}C_1t=C_2\rangle</math> is an [[HNN-extension]] where ''C''<sub>1</sub>, ''C''<sub>2</sub> are isomorphic finite [[subgroup]]s of ''H'' then ''G'' is a [[finitely generated group]] and ''e''(''G'') > 1.
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| Stallings' theorem shows that the converse is also true.
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| ==Formal statement of Stallings' theorem==
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| Let ''G'' be a [[finitely generated group]].
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| Then ''e''(''G'') > 1 if and only if one of the following holds:
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| *The group ''G'' admits a splitting ''G''=''H''∗<sub>''C''</sub>''K'' as a [[free product with amalgamation]] where ''C'' is a finite group such that ''C'' ≠ ''H'' and ''C'' ≠ ''K''.
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| *The group ''G'' admits a splitting <math>\scriptstyle G=\langle H, t| t^{-1}C_1t=C_2\rangle</math> is an [[HNN-extension]] where and ''C''<sub>1</sub>, ''C''<sub>2</sub> are isomorphic finite [[subgroup]]s of ''H''.
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| In the language of [[Bass-Serre theory]] this result can be restated as follows:
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| For a [[finitely generated group]] ''G'' we have ''e''(''G'') > 1 if and only if ''G'' admits a nontrivial (that is, without a global fixed vertex) [[group action|action]] on a simplicial [[tree (graph theory)|tree]] with finite edge-stabilizers and without edge-inversions.
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| For the case where ''G'' is a torsion-free [[finitely generated group]], Stallings' theorem implies that ''e''(''G'') = ∞ if and only if ''G'' admits a proper [[free product]] decomposition ''G'' = ''A''∗''B'' with both ''A'' and ''B'' nontrivial.
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| ==Applications and generalizations==
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| *Among the immediate applications of Stallings' theorem was a proof by Stallings<ref>John R. Stallings. [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183529548 ''Groups of dimension 1 are locally free.''] Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 361–364</ref> of a long-standing conjecture that every finitely generated group of cohomological dimension one is free and that every torsion-free [[virtually]] [[free group]] is free.
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| *Stallings' theorem also implies that the property of having a nontrivial splitting over a finite subgroup is a [[quasi-isometry]] invariant of a [[finitely generated group]] since the number of ends of a finitely generated group is easily seen to be a quasi-isometry invariant. For this reason Stallings' theorem is considered to be one of the first results in [[geometric group theory]].
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| *Stallings' theorem was a starting point for Dunwoody's ''accessibility theory''. A finitely generated group ''G'' is said to be ''accessible'' if the process of iterated nontrivial splitting of ''G'' over finite subgroups always terminates in a finite number of steps. In [[Bass-Serre theory]] terms that the number of edges in a reduced splitting of ''G'' as the fundamental group of a [[graph of groups]] with finite edge groups is bounded by some constant depending on ''G''. [[Martin Dunwoody|Dunwoody]] proved<ref name="D">M. J. Dunwoody. [http://www.springerlink.com/content/n13t047466x2g242/ ''The accessibility of finitely presented groups.''] [[Inventiones Mathematicae]], vol. 81 (1985), no. 3, pp. 449-457</ref> that every [[finitely presented group]] is accessible but that there do exist [[finitely generated group]]s that are not accessible.<ref>M. J. Dunwoody. ''An inaccessible group''. Geometric group theory, Vol. 1 (Sussex, 1991), pp. 75–78, London Mathematical Society Lecture Note Series, vol. 181, [[Cambridge University Press]], Cambridge, 1993; ISBN 0-521-43529-3</ref> Linnell<ref>P. A. Linnell. [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0K-45FC39Y-3Y&_user=10&_coverDate=10%2F31%2F1983&_rdoc=5&_fmt=high&_orig=browse&_srch=doc-info(%23toc%235649%231983%23999699998%23295444%23FLP%23display%23Volume)&_cdi=5649&_sort=d&_docanchor=&_ct=10&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=8e943c2dc0b8741d1f1a5bfc2cb0c1fd ''On accessibility of groups.'']
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| Journal of Pure and Applied Algebra, vol. 30 (1983), no. 1, pp. 39–46.</ref> showed that if one bounds the size of finite subgroups over which the splittings are taken then every finitely generated group is accessible in this sense as well. These results in turn gave rise to other versions of accessibility such as [[Mladen Bestvina|Bestvina]]-Feighn accessibility<ref>M. Bestvina and M. Feighn. [http://www.springerlink.com/content/t2321u08133g0475/ ''Bounding the complexity of simplicial group actions on trees.''] [[Inventiones Mathematicae]], vol. 103 (1991), no. 3, pp. 449–469</ref> of finitely presented groups (where the so-called "small" splittings are considered), acylindrical accessibility,<ref>Z. Sela. [http://www.springerlink.com/content/wmptp9tx4jr660x1/ ''Acylindrical accessibility for groups.''] [[Inventiones Mathematicae]], vol. 129 (1997), no. 3, pp. 527–565</ref><ref>T. Delzant. [http://www.numdam.org/numdam-bin/fitem?id=AIF_1999__49_4_1215_0 ''Sur l'accessibilité acylindrique des groupes de présentation finie.''] Université de Grenoble. Annales de l'Institut Fourier, vol. 49 (1999), no. 4, pp. 1215–1224</ref> strong accessibility,<ref>T. Delzant, and L. Potyagailo. [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1J-41KP586-B&_user=10&_coverDate=05%2F31%2F2001&_rdoc=10&_fmt=high&_orig=browse&_srch=doc-info(%23toc%235676%232001%23999599996%23216973%23FLA%23display%23Volume)&_cdi=5676&_sort=d&_docanchor=&_ct=11&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=26b77dd7de7faf457984fc962a78f4f6 ''Accessibilité hiérarchique des groupes de présentation finie''.] [[Topology (journal)|Topology]], vol. 40 (2001), no. 3, pp. 617–629</ref> and others.
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| *Stallings' theorem is a key tool in proving that a finitely generated group ''G'' is [[virtually]] [[free group|free]] if and only if ''G'' can be represented as the fundamental group of a finite [[graph of groups]] where all vertex and edge groups are finite (see, for example,<ref>H. Bass. [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0K-4RFD0C8-1&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=8743e9ee8227c380b4c799b5631b6c38 ''Covering theory for graphs of groups.''] Journal of Pure and Applied Algebra, vol. 89 (1993), no. 1-2, pp. 3–47</ref>).
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| *Using Dunwoody's accessibility result, Stallings' theorem about ends of groups and the fact that if G is a finitely presented group with asymptotic dimension 1 then G is virtually free<ref name="Gen"/> one can show <ref name="Gr"/> that for a finitely presented [[word-hyperbolic group]] ''G'' the hyperbolic boundary of ''G'' has [[topological dimension]] zero if and only if ''G'' is virtually free.
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| *Relative versions of Stallings' theorem and relative ends of [[finitely generated group]]s with respect to subgroups have also been considered. For a subgroup ''H''≤''G'' of a finitely generated group ''G'' one defines ''the number of relative ends'' ''e''(''G'',''H'') as the number of ends of the relative Cayley graph (the [[Cayley graph#Schreier coset graph|Schreier coset graph]]) of ''G'' with respect to ''H''. The case where ''e''(''G'',''H'')>1 is called a semi-splitting of ''G'' over ''H''. Early work on semi-splittings, inspired by Stallings' theorem, was done in the 1970s and 1980s by Scott,<ref>Peter Scott. [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0K-45FT7S1-2S&_user=10&_coverDate=12%2F31%2F1977&_rdoc=20&_fmt=high&_orig=browse&_srch=doc-info(%23toc%235649%231977%23999889998%23298178%23FLP%23display%23Volume)&_cdi=5649&_sort=d&_docanchor=&_ct=28&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=6b73e942917fbe66734387c5551218ab ''Ends of pairs of groups.'']
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| Journal of Pure and Applied Algebra, vol. 11 (1977/78), no. 1–3, pp. 179–198</ref> Swarup,<ref>G. A. Swarup. [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0K-45FT7S1-2F&_user=10&_coverDate=12%2F31%2F1977&_rdoc=11&_fmt=high&_orig=browse&_srch=doc-info(%23toc%235649%231977%23999889998%23298178%23FLP%23display%23Volume)&_cdi=5649&_sort=d&_docanchor=&_ct=28&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=313f2d84abfb5843e125467c44e82bd0 ''Relative version of a theorem of Stallings.'']
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| Journal of Pure and Applied Algebra, vol. 11 (1977/78), no. 1–3, pp. 75–82</ref> and others.<ref>H. Müller. [http://www.springerlink.com/content/j3038x25847q8455/ ''Decomposition theorems for group pairs.'']
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| Mathematische Zeitschrift, vol. 176 (1981), no. 2, pp. 223–246</ref><ref>P. H. Kropholler, and M. A. Roller. [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0K-45D9SSN-11&_user=10&_coverDate=11%2F12%2F1989&_rdoc=7&_fmt=high&_orig=browse&_srch=doc-info(%23toc%235649%231989%23999389997%23291829%23FLP%23display%23Volume)&_cdi=5649&_sort=d&_docanchor=&_ct=7&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=d7ff7c65ddee3b751b333adf1cb17df6 ''Relative ends and duality groups''.] Journal of Pure and Applied Algebra, vol. 61 (1989), no. 2, pp. 197–210</ref> The work of Sageev<ref>Michah Sageev. [http://plms.oxfordjournals.org/cgi/content/abstract/s3-71/3/585 ''Ends of group pairs and non-positively curved cube complexes.''] Proceedings of the London Mathematical Society (3), vol. 71 (1995), no. 3, pp. 585–617</ref> and Gerasomov<ref>V. N. Gerasimov. ''Semi-splittings of groups and actions on cubings.'' (in Russian) Algebra, geometry, analysis and mathematical physics (Novosibirsk, 1996), pp. 91–109, 190, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 1997</ref> in the 1990s showed that for a subgroup ''H''≤''G'' the condition ''e''(''G'',''H'')>1 correpsonds to the group ''G'' admitting an essential isometric action on a [[CAT(0) space|CAT(0)-cubing]] where a subgroup commensurable with ''H'' stabilizes an essential "hyperplane" (a simplicial tree is an example of a CAT(0)-cubing where the hyperplanes are the midpoints of edges). In certain situations such a semi-splitting can be promoted to an actual algebraic splitting, typically over a subgroup commensurable with ''H'', such as for the case where ''H'' is finite (Stallings' theorem). Another situation where an actual splitting can be obtained (modulo a few exceptions) is for semi-splittings over virtually [[polycyclic group|polycyclic]] subgroups. Here the case of semi-splittings of [[word-hyperbolic group]]s over two-ended (virtually infinite cyclic) subgroups was treated by Scott-Swarup<ref>G. P. Scott, and G. A. Swarup. [http://nyjm.albany.edu:8000/PacJ/p/2000/196-2-13.pdf ''An algebraic annulus theorem.''] Pacific Journal of Mathematics, vol. 196 (2000), no. 2, pp. 461–506</ref> and by [[Brian Bowditch|Bowditch]].<ref>B. H. Bowditch. [http://www.springerlink.com/content/a32707388262615w/?p=a1c491d95ab5404fa10c50e2ca306ee6&pi=0 ''Cut points and canonical splittings of hyperbolic groups.'']
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| [[Acta Mathematica]], vol. 180 (1998), no. 2, pp. 145–186</ref> The case of semi-splittings of [[finitely generated group]]s with respect to virtually polycyclic subgroups is dealt with by the algebraic torus theorem of Dunwoody-Swenson.<ref>M. J. Dunwoody, and E. L. Swenson. [http://www.springerlink.com/content/892hyejtew7h2vg5/ ''The algebraic torus theorem.''] [[Inventiones Mathematicae]], vol. 140 (2000), no. 3, pp. 605–637</ref>
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| *A number of new proofs of Stallings' theorem have been obtained by others after Stallings' original proof. [[Martin Dunwoody|Dunwoody]] gave a proof<ref>M. J. Dunwoody. [http://www.springerlink.com/content/yp22n46n40813lwr/ ''Cutting up graphs.'']
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| Combinatorica, vol. 2 (1982), no. 1, pp. 15–23</ref> based on the ideas of edge-cuts. Later Dunwoody also gave a proof of Stallings' theorem for finitely presented groups using the method of "tracks" on finite 2-complexes.<ref name="D"/> Niblo obtained a proof<ref>Graham A. Niblo. [http://www.springerlink.com/content/l872767m33653476/ ''A geometric proof of Stallings' theorem on groups with more than one end.'']
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| [[Geometriae Dedicata]], vol. 105 (2004), pp. 61–76</ref> of Stallings' theorem as a consequence of Sageev's CAT(0)-cubing relative version, where the CAT(0)-cubing is eventually promoted to being a tree. Niblo's paper also defines an abstract group-theoretic obstruction (which is a union of double cosets of ''H'' in ''G'') for obtaining an actual splitting from a semi-splitting. It is also possible to prove Stallings' theorem for [[finitely presented group]]s using [[Riemannian geometry]] techniques of [[minimal surface]]s, where one first realizes a finitely presented group as the fundamental group of a compact 4-manifold (see, for example, a sketch of this argument in the survey article of [[Terry Wall|Wall]]<ref>C. T. C. Wall. ''The geometry of abstract groups and their splittings.'' Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5–101</ref>). [[Mikhail Gromov (mathematician)|Gromov]] outlined a proof (see pp. 228–230 in <ref name="Gr">M. Gromov, Hyperbolic Groups, in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75-263</ref>) where the minimal surfaces argument is replaced by an easier harmonic analysis argument and this approach was pushed further by Kapovich to cover the original case of finitely generated groups.<ref name ="Gen">Gentimis Thanos, Asymptotic dimension of finitely presented groups, http://www.ams.org/journals/proc/2008-136-12/S0002-9939-08-08973-9/home.html</ref><ref>M. Kapovich. [http://arxiv.org/abs/0707.4231 ''Energy of harmonic functions and Gromov's proof of Stallings' theorem''], preprint, 2007, arXiv:0707.4231</ref>
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| ==See also==
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| *[[Free product with amalgamation]]
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| *[[HNN extension]]
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| *[[Bass-Serre theory]]
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| *[[Graph of groups]]
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| *[[Geometric group theory]]
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| ==Notes==
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| {{reflist}}
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| {{DEFAULTSORT:Stallings Theorem About Ends Of Groups}}
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| [[Category:Geometric group theory]]
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| [[Category:Theorems in group theory]]
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