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APPEARED IN BULLETIN OF THE

arXiv:math/9201264v1 [math.GR] 1 Jan 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 1, Jan 1992, Pages 131-136SEMISTABILITY OF AMALGAMATED PRODUCTS,HNN-EXTENSIONS, AND ALL ONE-RELATOR GROUPSMichael L. Mihalik and Steven T. Tschantz1. IntroductionSemistability at infinity is a geometric property used in the study of ends offinitely presented groups.

If a finitely presented group G is semistable at infinity,then sophisticated invariants for G, such as the fundamental group at an end of G,can be defined (see [10]). It is unknown whether or not all finitely presented groupsare semistable at infinity, although by [16] it suffices to know whether all 1-endedfinitely presented groups are semistable at infinity.

There are a number of resultsshowing many 1-ended groups have this property, e.g., if G is finitely presented andcontains a finitely generated, infinite, normal subgroup of infinite index, then G issemistable at infinity (see [12] and for other such results [13–15]).Semistability at infinity is of interest in the study of cohomology of groups; if afinitely presented group G is semistable at infinity, then H2(G; ZG) is free abelian(see [6, 7]). This is conjectured to be true for all finitely presented groups, butat present it is not even known for 2-dimensional duality groups (where one isdiscussing the dualizing module, see [2]).For negatively curved groups (i.e., hyperbolic groups in the sense of Gromov, see[8]), semistability at infinity has additional interesting consequences.

If a negativelycurved group G is given the word metric with respect to some finite generating set,then there is a compactification G of G where a point of ∂G = G −G is a certainequivalence class of proper sequences of points in G.The boundary of G is acompact, metrizable, finite-dimensional space, which determines the cohomology ofG. Bestvina and Mess have shown that if G is a negatively curved group, then forevery ring R, there is an isomorphism of RG-modules Hi(G; RG) ∼= ˇHi−1(∂G; R)(ˇCech reduced).

Geoghegan has observed that results in [1] imply that a negativelycurved group G is semistable at infinity iff∂G has the shape of a locally connectedcontinuum (see [6]). Furthermore, in [1], ideas closely related to semistability atinfinity are used to analyze closed irreducible 3-manifolds with negatively curvedfundamental group.A continuous map is proper if inverse images of compact sets are compact.

Properrays r, s: [0, ∞) →K in a locally finite CW-complex K are said to converge to thesame end of K if for every compact C ⊆K there exists an N such that r([N, ∞))and s([N, ∞)) are contained in the same path component of K −C. A locally finite1991 Mathematics Subject Classification.

Primary 20F32; Secondary 20E06, 57M20.Key words and phrases. Semistability at infinity, amalgamated products, HNN-extensions,one-relator groups, group cohomology, finitely presented groups, proper homotopies.Received by the editors August 29, 1990 and, in revised form, June 10, 1991c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2M. L. MIHALIK AND S. T. TSCHANTZCW-complex K is semistable at infinity if any two proper rays, which converge tothe same end of K, are properly homotopic.

If G is a finitely presented group, thenG is semistable at infinity if for some (equivalently any) finite CW-complex X withπ1(X) = G, the universal cover eX of X is semistable at infinity.If H is a subgroup of two groups A and B, the amalgamated product A ∗H B isthe quotient of the free product of A and B where the copies of H in A and B areidentified. If H and H′ are isomorphic subgroups of A, the HNN-extension A∗H(where H′ is taken as given) is the quotient of A ∗⟨t⟩where H is identified withtH′t−1 (see [11]).

In [21], Stallings proves a decomposition theorem for finitelygenerated groups having more than one end in terms of amalgamated productsor HNN-extensions over finite subgroups. In [4], Dunwoody shows that for finitelypresented groups, the process of recursively applying this decomposition theorem tothe factor groups eventually terminates in 0-ended (i.e., finite) and 1-ended factorgroups.

Our main result is the following:Theorem 1. If G = A ∗H B is an amalgamated product where A and B arefinitely presented and semistable at infinity, and H is finitely generated, then G issemistable at infinity.

If G = A∗H is an HNN-extension where A is finitely pre-sented and semistable at infinity, and H is finitely generated, then G is semistableat infinity.If G is the fundamental group of a graph of groups (see [20]), then G can beexpressed as some combination of amalgamated products and HNN-extensions ofthe vertex groups over the edge groups. Hence, if G is the fundamental group of afinite graph of groups in which each vertex group is finitely presented and semistableat infinity and each edge group is finitely generated, then G is semistable at in-finity.

However, it is possible that a group G can be expressed as a combinationof amalgamated products and HNN-extensions of finitely presented groups overfinitely generated (but not finite) groups without G being the fundamental groupof a graph of groups with these vertex and edge groups, hence the above theoremapplies to a larger class of group decomposition. Although the question of semista-bility at infinity for all finitely presented groups reduces to the same question for1-ended groups, it is possible to obtain a 1-ended group G = A ∗H B where A, B,and H are infinite-ended (and similarly for HNN-extensions), and in fact this is theessential difficulty in the proof of our main theorem.As a corollary to the proof of Theorem 1, the same methods apply (with homo-topy replaced by homology in the sense of [7]) to give a cohomology version of thisresult.Corollary 2.

If G = A ∗H B is an amalgamated product where A and B arefinitely presented, H2(A, ZA) and H2(B; ZB) are free abelian, and H is finitelygenerated, then H2(G; ZG) is free abelian. If G = A∗H is an HNN-extension whereA is finitely presented, H2(A; ZA) is free abelian, and H is finitely generated, thenH2(G; ZG) is free abelian.As an application of our main result, we get the following general theorem:Theorem 3.

All finitely generated one-relator groups are semistable at infinity.Finally, as a corollary (using [7] as before), we get a purely cohomological result.Corollary 4. If G is a finitely generated one-relator group, then H2(G; ZG) is freeabelian.

AMALGAMATED PRODUCTS, HNN-EXTENSIONS, ONE-RELATOR GROUPS3Figure 12. Outline of proofsWe describe the proof of our main theorem in the amalgamated product case.Take a presentation P for G = A ∗H B by combining presentations for A and B,each containing generators for H. If Z is the standard 2-complex obtained fromP, then Z = X ∪Y where X and Y are subcomplexes of Z with π1(X) = A andπ1(Y ) = B, and X ∩Y is a wedge of circles representing generators for H in bothπ1(X) and π1(Y ).

The universal cover eZ of Z is a union of copies of eX and eYattached along copies of the Cayley graph Γ of H. The group G acts on the left ofeZ, permuting copies of eX, eY , and Γ.To prove the main theorem, we show that any two proper edge paths r and sin eZ, converging to the same end of eZ, are properly homotopic. The normal formstructure of A ∗H B provides the geometric structure to show that r and s areproperly homotopic in case im(r) ∪im(s) intersects no copy of Γ in an infinite setof vertices.If r or s meets some copy of Γ, say Γ0, in infinitely many vertices, then (byreplacing each ray with a properly homotopic ray passing through these points) wemay as well assume V = im(r) ∩im(s) ∩Γ0 contains infinitely many vertices.

Letq be a proper edge path in Γ0 passing through infinitely many vertices in V . Thenq and r (and s) converge to the same end of eZ, and it suffices to show that q and rare properly homotopic (since then q and s are similarly properly homotopic, andthus r and s are properly homotopic).

Thus we are reduced to the case where oneof our rays is contained in a copy Γ0 of Γ.The main ideas in this, the main case in our work, are as follows. We spliteZ into two connected pieces eZ+ and eZ−, which intersect along Γ0 by taking eX0and eY0 to be the copies of eX and eY containing Γ0 and then defining eZ+ to bethe component of ( eZ −eY0) ∪Γ0 containing Γ0, and eZ−to be the component of( eZ −eX0) ∪Γ0 containing Γ0.

By extracting ideas from the proof of Dunwoody’saccessibility theorem [4], we show that a certain configuration of rays and endscannot occur in eZ+ or eZ−. (This configuration is represented in Figure 1, where Cis a compact set in eZ; u, v, u′i, and v′i are proper rays in Γ0, with the u′i and v′i indifferent components of Γ0 −C and diverging from u and v at progressively laterpoints, and where ovals represent distinct ends of either eZ+ or eZ−.) Because thisconfiguration cannot occur, we can constructproper homotopies between any proper ray in Γ0 and any proper ray in eZ+ or( eZ−) that converge to the same end of eZ+ (respectively, eZ−).

In essence, this saysthat the ends of eZ+ and eZ−, determined by Γ0, are semistable at infinity. This

4M. L. MIHALIK AND S. T. TSCHANTZfact provides the geometric structure needed to construct a patchwork of properhomotopies in eZ, giving a proper homotopy between r and q and, thus, betweenthe given r and s.The proof that all one-relator groups are semistable at infinity is by an inductionargument patterned after the proof by Magnus of the Freiheitssatz (see [11]).

Theproof makes use of our main theorem, the following structure theorem for one-relator groups, and a simple fact about semistability at infinity for factor groups incertain amalgamated products.Lemma 5. Given any finitely generated one relator group G, there exists a finitesequence of finitely generated one relator groups H1, H2, .

. .

, Hn = G such that,for each i < n, either Hi+1 or Hi+1 ∗Z is an HNN-extension of Hi over a finitelygenerated group, and such that H1 is either a free group or else is isomorphic to afree product of a free group and a finite cyclic group.Lemma 6. If G is finitely presented and G ∗Z is semistable at infinity, then G issemistable at infinity.References1.

M. Bestvina and G. Mess, The boundary of negatively curved groups, preprint, 1990.2. R. Bieri, Homological dimension of discrete groups, Queen Mary College Math.

Notes,London, 1976.3. W. Dicks and M. J. Dunwoody, Groups acting on graphs, Cambridge Univ.

Press, Cam-bridge and New York, 1989.4. M. J. Dunwoody, The accessibility of finitely presented groups, Invent.

Math. 81 (1985),449–457.5.

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33 (1931),692–713.6. R. Geoghegan, The shape of a group, Geometric and Algebraic Topology (H.

Torunczyk,ed. ), Banach Center Publ., vol.

18, PWN, Warsaw, 1986, pp. 271–280.7.

R. Geoghegan and M. Mihalik, Free abelian cohomology of groups and ends of universalcovers, J. Pure Appl.

Algebra 36 (1985), 123–137.8. M. Gromov, Hyperbolic Groups, Essays in Group Theory (S. M. Gersten, ed.

), Math. Sci.Res.

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75–263.9. H. Hopf, Enden offener Ra¨ume und unendliche diskontinuierliche Gruppen, Comment.Math.

Helv 16 (1943), 81–100.10. B. Jackson, End invariants of groups extensions, Topology 21 (1982), 71–81.11.

R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin,Heidelberg and New York, 1970.12. M. Mihalik, Semistability at the end of a group extension, Trans.

Amer. Math.

Soc. 277(1983), 307–321.13., Ends of groups with the integers as quotients, J.

Pure Appl. Algebra 35 (1985),305–320.14., Ends of double extension groups, Topology 25 (1986), 45–53.15., Semistability at ∞of finitely generated groups and solvable groups, TopologyAppl.

24 (1986), 259–269.16., Semistability at ∞, ∞-ended groups and group cohomology, Trans. Amer.

Math.Soc. 303 (1987), 479–485.17.

M. Mihalik and S. Tschantz, Semistability of amalgamated products and HNN-extensions,Mem. Amer.

Math. Soc.

(to appear).18., All one relator groups are semistable at infinity, preprint, 1991.19. G. P. Scott and T. Wall, Topological methods in group theory, London Math.

Soc. LectureNote Ser., vol.

36, Cambridge Univ. Press, 1979, pp.

137–203.20. J.-P. Serre, Trees, Springer-Verlag, Berlin and New York, 1980.

AMALGAMATED PRODUCTS, HNN-EXTENSIONS, ONE-RELATOR GROUPS521. J. Stallings, Group theory and three dimensional manifolds, Yale Math.

Mono., vol. 4, YaleUniv.

Press, New Haven, CT, 1972.Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37235

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