# Modern references on hyperbolic groups

Several good references dedicated to hyperbolic groups have been written until 1990, including:

• Hyperbolic groups, written by M. Gromov.

• Géométrie et théorie des groupes : les groupes hyperboliques de Gromov, written by M. Coornaert, A. Papadopoulos and T. Delzant.

• Sur les groupes hyperboliques de M. Gromov, edited by E. Ghys and P. de la Harpe.

Since then, fundamental tools have been introduced to study hyperbolic groups. For instance, I have in mind JSJ decompositions. Are there textbooks or surveys on the subject?

What are good references dedicated to modern developments on hyperbolic groups?

By "modern", I mean subjects not contained in the references mentioned above. I am particularly interested in progresses made in the study of outer automorphism groups of hyperbolic groups.

Edit: I am aware that the ideas involved in the study of hyperbolic groups have lots of applications, so that covering them in just a few references is hopeless. Consequently, it would be better to focus on hyperbolic themselves, excluding generalisations (such as relatively / acylindrically hyperbolic groups) or specific examples (such as right-angled Coxeter groups, (automorphisms of) free groups or (automorphisms of) surface groups).

• There's this 2002 survey by I. Kapovich and Benakli, specifically about boundaries of hyperbolic groups: arxiv.org/abs/math/0202286 – YCor Aug 24 '18 at 7:47
• Gromov's hyperbolic groups has been published in 1987, probably written in 85-86. Both other books have been published in 1990, so these rather provide the state of art at the end of the 80's. – YCor Aug 24 '18 at 7:48
• I'm not sure this is what you are looking for, so this is just a comment and not an answer: for things related to ergodic theory in hyperbolic groups, you have a very nice modern survey of Danny Calegari: arxiv.org/abs/1111.0029 – M. Dus Aug 24 '18 at 9:28
• – mo-user Aug 24 '18 at 10:15
• The theory of hyperbolic groups with applications to numerous other areas is too large now for a book that starts with basics and covers all that is known.Just an example, it should include Agol-Wise results. And Sela-Kharlampovich-Myasnikov results. And relatively hyperbolic groups. I doubt such a series of books will ever be written. . – user6976 Aug 24 '18 at 14:15

I think this is a great question, as there is still a need for an authoritative reference about (word-)hyperbolic groups. Since the textbook doesn't exist, I'd like to take the question in a slightly different direction by listing some of the material I think it should cover. (This is inevitably a personal and biased account.) I'll try to include the best references I can think of.

The seminal article of Gromov's mentioned in the question contained many assertions about hyperbolic groups, often extensions of Thurston's famous theorems about hyperbolic manifolds. Probably the simplest such statement (which is fundamental in the study of outer automorphism groups of free groups) is now known as Paulin's theorem (cf. 5.4.A of Hyperbolic groups).

Paulin's theorem: If $$\Gamma$$ is a torsion-free hyperbolic group that doesn't split over a cyclic subgroup then $$\mathrm{Out}(\Gamma)$$ is finite.

Another such statement is 5.3.C' from Gromov's article.

Subgroup Rigidity Theorem: Let $$\Gamma$$ be a hyperbolic group and $$H$$ a one-ended finitely presented group. Then there are only finitely many conjugacy classes of subgroups of $$\Gamma$$ isomorphic to $$H$$.

In Hyperbolic groups, Gromov suggests that statements like these can be proved using a generalisation of Thurston's arguments using the geodesic flow. Even the problem of constructing a candidate geodesic flow over a hyperbolic group is notoriously difficult. The problem was eventually solved by Mineyev but, to the best of my knowledge, no one so far been able to use Mineyev's geodesic flow to give the Thurstonian proofs of these result that Gromov suggested.

Instead, the key tool in proving these results turned out to be the Rips machine: Rips' classification of certain actions of groups on real trees. For actions of finitely presented groups, the Rips machine was developed by Bestvina—Feighn. An account was also given by Misha Kapovich in his book, so the interested reader could look at either of the following.

• Bestvina & Feighn, Stable actions of groups on real trees. Invent. Math. 121 (1995), no. 2, 287–321
• M. Kapovich, Hyperbolic manifolds and discrete groups. Progress in Mathematics, 183. Birkhäuser Boston, Inc., Boston, MA, 2001. xxvi+467 pp

For some applications, one needs the Rips machine for finitely generated groups. This was developed initially by Sela and corrected and refined by Guirardel, so the relevant references here are:

• Sela, Acylindrical accessibility for groups. Invent. Math. 129 (1997), no. 3, 527–565.
• Guirardel, Actions of finitely generated groups on $$\mathbb{R}$$-trees. Ann. Inst. Fourier (Grenoble) 58 (2008), no. 1, 159–211.

The link with hyperbolic groups is made via what Sela called the Bestvina—Paulin method. Given infinitely many actions of a group $$G$$ on a suitably nice $$\delta$$-hyperbolic space $$X$$, one can pass to a limiting action on a space which is either $$X$$ itself or a real tree $$T$$, and in the latter case Rips’ machine applies. As well as Paulin’s original paper, another account of the proof, also modulo the Rips machine, was given by Bridson—Swarup (who corrected a small mistake in Paulin’s proof), and Bestvina gave a very useful account in his survey article on $$\mathbb{R}$$-trees. So one could look at:

• Paulin, Outer automorphisms of hyperbolic groups and small actions on $$\mathbb{R}$$-trees., Arboreal group theory (Berkeley, CA, 1988), 331–343, Math. Sci. Res. Inst. Publ., 19, Springer, 1991.
• Bridson & Swarup, On Hausdorff-Gromov convergence and a theorem of Paulin, Enseign. Math. (2) 40 (1994), no. 3-4, 267–289.
• Bestvina, $$\mathbb{R}$$-trees in topology, geometry, and group theory. Handbook of geometric topology, 55–91, North-Holland, Amsterdam, 2002.

For Paulin’s theorem, the only thing one needs from the Rips machine is that it promotes a (nice) action on a real tree to a (nice) action on a simplicial tree. Deeper applications, such as the Subgroup Rigidity Theorem, tend to require a notorious trick called the shortening argument. The idea is that if the actions of $$G$$ on $$X$$ were all chosen to be `shortest’ in their conjugacy classes then either $$G$$ is a free product or the limiting action of $$G$$ on $$T$$ isn’t faithful. This trick is notoriously, er, tricky.

The first reference is Rips—Sela’s original paper in which they prove the Subgroup Rigidity Theorem in the torsion-free case. (The case with torsion was later handled by Delzant.)

• Rips & Sela, Structure and rigidity in hyperbolic groups. I. GAFA, 1994.

The shortening argument is Theorem 4.3, and the Subgroup Rigidity Theorem is Theorem 7.1. I gave an account of the shortening argument in Theorem 5.1 of

• Wilton, Solutions to Bestvina and Feighn's exercises on limit groups, Geometric and cohomological methods in group theory, pp. 30–62, LMS Lect. Note Ser. 358, CUP, 2009

Another account of the shortening argument (and many of the theorems mentioned here) adapted to the setting of toral relatively hyperbolic groups was given by Groves in:

• Groves, Limit groups for relatively hyperbolic groups. II. Makanin-Razborov diagrams. Geom. Topol. 9 (2005), 2319–2358.

This toolkit has some further spectacular consequences for the structure of hyperbolic groups. The two biggest are probably the Hopf property and the isomorphism problem.

Recall that a group $$G$$ is said to be non-Hopfian if there is a non-injective epimorphism $$G\to G$$. In

• Sela, Endomorphisms of hyperbolic groups. I. The Hopf property. Topology 38 (1999), no. 2, 301–321.

Sela proved the Hopf property for all torsion-free hyperbolic groups. The case with torsion is treated in a preprint of Reinfeldt—Weidmann.

Sela solved the isomorphism problem for torsion-free rigid hyperbolic groups (such as the hyperbolic 3-manifold groups) in

• Sela, The isomorphism problem for hyperbolic groups. I. Ann. of Math. (2) 141 (1995), no. 2, 217–283.

To extend this to the non-rigid case, one needs JSJ theory for groups, which was introduced by Rips—Sela in

• Rips & Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition. Ann. of Math. (2), 146 (1997), no. 1, 53–109.

As is already apparent from Ian’s answer, this theory has been developed a great deal, and the more recent approaches are frankly a lot simpler than the original Rips—Sela version. In the end, the torsion-free non-rigid case (as well as the toral relatively hyperbolic case) was dealt with in

• Dahmani & Groves, The isomorphism problem for toral relatively hyperbolic groups. Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 211–290.

while the case with torsion was dealt with in

• Dahmani & Guirardel, The isomorphism problem for all hyperbolic groups. Geom. Funct. Anal. 21 (2011), no. 2, 223–300.

Both of these papers followed Sela’s outline, although major technicalities needed to be overcome.

NWMT has alreay said in comments that all this is too difficult for a textbook. In total this is true, but I can imagine an advanced reference book that describes the Rips machine, the Bestvina—Paulin method and the shortening argument, and gives Paulin’s theorem and the Subgroup Rigidity theorem as applications. These are still some of the most amazing and beautiful arguments in the theory of hyperbolic groups, and I’m a little concerned that a lot of current research seems to be moving away from them, rather than attempting to simplify or extend them.

For JSJ decompositions of hyperbolic groups, I'm not sure there's much more that is known than the classical results on this topic. Most progress recently has been algorithmic - a new approach is given here by Benjamin Barrett.

The most intricate outer automorphism groups of hyperbolic groups are mapping class groups of surfaces and $Out(F_n)$. These are huge topics of current research. Indeed, all outer automorphism groups of one-ended hyperbolic groups are built out of these and finite groups. Here are some references:

Farb, Benson (ed.), Problems on mapping class groups and related topics, Proceedings of Symposia in Pure Mathematics 74. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3838-5/hbk). x, 371 p. (2006). ZBL1098.58001.

Farb, Benson; Margalit, Dan, A primer on mapping class groups, Princeton Mathematical Series. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/ebook). xiv, 492 p. (2011). ZBL1245.57002.

For the general theory of JSJ splittings, see:

Guirardel, Vincent; Levitt, Gilbert, JSJ decompositions of groups, Astérisque 395. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-870-1/pbk). vii, 165 p. (2017). ZBL1391.20002.

• Perhaps Bestvina's survey on R-trees in the handbook of Geometric Topology should be included as well. – NWMT Aug 25 '18 at 14:24