# Quasi-actions on trees II: Finite depth Bass-Serre trees

### About this Title

**Lee Mosher**, *Rutgers University, Newark*, **Michah Sageev**, *Technion, Israel University of Technology* and **Kevin Whyte**, *University of Illinois at Chicago*

Publication: Memoirs of the American Mathematical Society

Publication Year:
2011; Volume 214, Number 1008

ISBNs: 978-0-8218-4712-1 (print); 978-1-4704-0625-7 (online)

DOI: http://dx.doi.org/10.1090/S0065-9266-2011-00585-X

Published electronically: April 1, 2011

MathSciNet review: 2867450

MSC: Primary 20F65

### Table of Contents

**Chapters**

- Chapter 1. Introduction
- Chapter 2. Preliminaries
- Chapter 3. Depth zero vertex rigidity
- Chapter 4. Finite depth graphs of groups
- Chapter 5. Tree rigidity
- Chapter 6. Main theorems
- Chapter 7. Applications and examples

### Abstract

This paper addresses questions of quasi-isometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the Bass-Serre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge groups are coarse Poincaré duality groups. The main theorem says that, under certain hypotheses, if $\mathcal{G}$ is a finite graph of coarse Poincaré duality groups, then any finitely generated group quasi-isometric to the fundamental group of $\mathcal{G}$ is also the fundamental group of a finite graph of coarse Poincaré duality groups, and any quasi-isometry between two such groups must coarsely preserve the vertex and edge spaces of their Bass-Serre trees of spaces. Besides some simple normalization hypotheses, the main hypothesis is the ``crossing graph condition'', which is imposed on each vertex group $\mathcal{G}_v$ which is an $n$-dimensional coarse Poincaré duality group for which every incident edge group has positive codimension: the crossing graph of $\mathcal{G}_v$ is a graph $\epsilon_v$ that describes the pattern in which the codimension 1 edge groups incident to $\mathcal{G}_v$ are crossed by other edge groups incident to $\mathcal{G}_v$, and the crossing graph condition requires that $\epsilon_v$ be connected or empty.

**[BB00]**N. Brady and M. R. Bridson,*There is only one gap in the isoperimetric spectrum*, Geom. Funct. Anal.**10**(2000), no. 5, 1053–1070. MR**1800063**, 10.1007/PL00001646**[BK90]**Hyman Bass and Ravi Kulkarni,*Uniform tree lattices*, J. Amer. Math. Soc.**3**(1990), no. 4, 843–902. MR**1065928**, 10.1090/S0894-0347-1990-1065928-2**[BK98]**D. Burago and B. Kleiner,*Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps*, Geom. Funct. Anal.**8**(1998), no. 2, 273–282. MR**1616135**, 10.1007/s000390050056**[Bro75]**Kenneth S. Brown,*Homological criteria for finiteness*, Comment. Math. Helv.**50**(1975), 129–135. MR**0376820****[Bro82]**Kenneth S. Brown,*Cohomology of groups*, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR**672956****[BW97]**Jonathan Block and Shmuel Weinberger,*Large scale homology theories and geometry*, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 522–569. MR**1470747****[Cas]**A. Casson,*Three-dimensional topology*, lecture notes.**[Cas07]**Christopher H. Cashen,*Quasi-isometries between tubular groups*, Groups Geom. Dyn.**4**(2010), no. 3, 473–516. MR**2653972**, 10.4171/GGD/92**[CC92]**J. W. Cannon and Daryl Cooper,*A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three*, Trans. Amer. Math. Soc.**330**(1992), no. 1, 419–431. MR**1036000**, 10.1090/S0002-9947-1992-1036000-0**[CK00]**Christopher B. Croke and Bruce Kleiner,*Spaces with nonpositive curvature and their ideal boundaries*, Topology**39**(2000), no. 3, 549–556. MR**1746908**, 10.1016/S0040-9383(99)00016-6**[Dav98]**Michael W. Davis,*The cohomology of a Coxeter group with group ring coefficients*, Duke Math. J.**91**(1998), no. 2, 297–314. MR**1600586**, 10.1215/S0012-7094-98-09113-X**[DS00]**M. J. Dunwoody and E. L. Swenson,*The algebraic torus theorem*, Invent. Math.**140**(2000), no. 3, 605–637. MR**1760752**, 10.1007/s002220000063**[Dun85]**M. J. Dunwoody,*The accessibility of finitely presented groups*, Invent. Math.**81**(1985), no. 3, 449–457. MR**807066**, 10.1007/BF01388581**[FM98]**Benson Farb and Lee Mosher,*A rigidity theorem for the solvable Baumslag-Solitar groups*, Invent. Math.**131**(1998), no. 2, 419–451. With an appendix by Daryl Cooper. MR**1608595**, 10.1007/s002220050210**[FM99]**Benson Farb and Lee Mosher,*Quasi-isometric rigidity for the solvable Baumslag-Solitar groups. II*, Invent. Math.**137**(1999), no. 3, 613–649. MR**1709862**, 10.1007/s002220050337**[FM00]**Benson Farb and Lee Mosher,*On the asymptotic geometry of abelian-by-cyclic groups*, Acta Math.**184**(2000), no. 2, 145–202. MR**1768110**, 10.1007/BF02392628**[FM02a]**Benson Farb and Lee Mosher,*Convex cocompact subgroups of mapping class groups*, Geom. Topol.**6**(2002), 91–152 (electronic). MR**1914566**, 10.2140/gt.2002.6.91**[FM02b]**B. Farb and L. Mosher,*The geometry of surface-by-free groups*, Geom. Funct. Anal.**12**(2002), no. 5, 915–963. MR**1937831**, 10.1007/PL00012650**[FS96]**Benson Farb and Richard Schwartz,*The large-scale geometry of Hilbert modular groups*, J. Differential Geom.**44**(1996), no. 3, 435–478. MR**1431001****[Ger93]**Steve M. Gersten,*Quasi-isometry invariance of cohomological dimension*, C. R. Acad. Sci. Paris Sér. I Math.**316**(1993), no. 5, 411–416 (English, with English and French summaries). MR**1209258****[GMRS98]**Rita Gitik, Mahan Mitra, Eliyahu Rips, and Michah Sageev,*Widths of subgroups*, Trans. Amer. Math. Soc.**350**(1998), no. 1, 321–329. MR**1389776**, 10.1090/S0002-9947-98-01792-9**[Gro82]**Michael Gromov,*Volume and bounded cohomology*, Inst. Hautes Études Sci. Publ. Math.**56**(1982), 5–99 (1983). MR**686042****[GW77]**Jack E. Graver and Mark E. Watkins,*Combinatorics with emphasis on the theory of graphs*, Springer-Verlag, New York-Berlin, 1977. Graduate Texts in Mathematics, Vol. 54. MR**0505525****[Hak61]**Wolfgang Haken,*Theorie der Normalflächen*, Acta Math.**105**(1961), 245–375 (German). MR**0141106****[Hig61]**G. Higman,*Subgroups of finitely presented groups*, Proc. Roy. Soc. Ser. A**262**(1961), 455–475. MR**0130286****[HV50]**Paul R. Halmos and Herbert E. Vaughan,*The marriage problem*, Amer. J. Math.**72**(1950), 214–215. MR**0033330****[JR88]**William Jaco and J. Hyam Rubinstein,*PL minimal surfaces in $3$-manifolds*, J. Differential Geom.**27**(1988), no. 3, 493–524. MR**940116****[Kap19]**M. Kapovich,*Lectures on geometric group theory*, preprint, HTTP://WWW.MATH. UCDAVIS.EDU/~KAPOVICH, 2006 August 19.**[KK99]**Michael Kapovich and Bruce Kleiner,*Coarse Alexander duality and duality groups*, J. Differential Geom.**69**(2005), no. 2, 279–352. MR**2168506****[KK00]**Michael Kapovich and Bruce Kleiner,*Hyperbolic groups with low-dimensional boundary*, Ann. Sci. École Norm. Sup. (4)**33**(2000), no. 5, 647–669 (English, with English and French summaries). MR**1834498**, 10.1016/S0012-9593(00)01049-1**[KL97]**Michael Kapovich and Bernhard Leeb,*Quasi-isometries preserve the geometric decomposition of Haken manifolds*, Invent. Math.**128**(1997), no. 2, 393–416. MR**1440310**, 10.1007/s002220050145**[Man05]**Jason Fox Manning,*Geometry of pseudocharacters*, Geom. Topol.**9**(2005), 1147–1185 (electronic). MR**2174263**, 10.2140/gt.2005.9.1147**[Mar81]**G. A. Margulis,*On the decomposition of discrete subgroups into amalgams*, Selecta Math. Soviet.**1**(1981), no. 2, 197–213. Selected translations. MR**672427****[McM98]**C. T. McMullen,*Lipschitz maps and nets in Euclidean space*, Geom. Funct. Anal.**8**(1998), no. 2, 304–314. MR**1616159**, 10.1007/s000390050058**[Mos97]**Lee Mosher,*A hyperbolic-by-hyperbolic hyperbolic group*, Proc. Amer. Math. Soc.**125**(1997), no. 12, 3447–3455. MR**1443845**, 10.1090/S0002-9939-97-04249-4**[MS04]**Nicolas Monod and Yehuda Shalom,*Cocycle superrigidity and bounded cohomology for negatively curved spaces*, J. Differential Geom.**67**(2004), no. 3, 395–455. MR**2153026****[MSW00]**L. Mosher, M. Sageev, and K. Whyte,*Quasi-actions on trees, research announcement*, preprint, ARXIV:MATH.GR/0005210, 2000.**[MSW02]**Lee Mosher, Michah Sageev, and Kevin Whyte,*Maximally symmetric trees*, Geom. Dedicata**92**(2002), 195–233. Dedicated to John Stallings on the occasion of his 65th birthday. MR**1934019**, 10.1023/A:1019685632755**[MSW03]**Lee Mosher, Michah Sageev, and Kevin Whyte,*Quasi-actions on trees. I. Bounded valence*, Ann. of Math. (2)**158**(2003), no. 1, 115–164. MR**1998479**, 10.4007/annals.2003.158.115**[MY82]**W. H. Meeks and S. T. Yau,*The equivariant Dehn's lemma and loop theorem*, Comment. Math. Helv.**56**(1982), 225-239.**[Nib04]**Graham A. Niblo,*A geometric proof of Stallings’ theorem on groups with more than one end*, Geom. Dedicata**105**(2004), 61–76. MR**2057244**, 10.1023/B:GEOM.0000024780.73453.e4**[Pap05]**Panos Papasoglu,*Quasi-isometry invariance of group splittings*, Ann. of Math. (2)**161**(2005), no. 2, 759–830. MR**2153400**, 10.4007/annals.2005.161.759**[Pap07]**Panos Papasoglu,*Group splittings and asymptotic topology*, J. Reine Angew. Math.**602**(2007), 1–16. MR**2300450**, 10.1515/CRELLE.2007.001**[PW02]**Panos Papasoglu and Kevin Whyte,*Quasi-isometries between groups with infinitely many ends*, Comment. Math. Helv.**77**(2002), no. 1, 133–144. MR**1898396**, 10.1007/s00014-002-8334-2**[RS94]**E. Rips and Z. Sela,*Structure and rigidity in hyperbolic groups. I*, Geom. Funct. Anal.**4**(1994), no. 3, 337–371. MR**1274119**, 10.1007/BF01896245**[Sch97]**Richard Evan Schwartz,*Symmetric patterns of geodesics and automorphisms of surface groups*, Invent. Math.**128**(1997), no. 1, 177–199. MR**1437498**, 10.1007/s002220050139**[Sel97]**Z. Sela,*Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank $1$ Lie groups. II*, Geom. Funct. Anal.**7**(1997), no. 3, 561–593. MR**1466338**, 10.1007/s000390050019**[Ser80]**Jean-Pierre Serre,*Trees*, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR**607504****[Sta68a]**John Stallings,*Groups of dimension 1 are locally free*, Bull. Amer. Math. Soc.**74**(1968), 361–364. MR**0223439**, 10.1090/S0002-9904-1968-11955-X**[Sta68b]**John R. Stallings,*On torsion-free groups with infinitely many ends*, Ann. of Math. (2)**88**(1968), 312–334. MR**0228573****[SW79]**Peter Scott and Terry Wall,*Topological methods in group theory*, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 137–203. MR**564422****[Wal65]**C. T. C. Wall,*Finiteness conditions for ${\rm CW}$-complexes*, Ann. of Math. (2)**81**(1965), 56–69. MR**0171284****[Why02]**K. Whyte,*The large scale geometry of the higher Baumslag-Solitar groups*, Geom. Funct. Anal.**11**(2001), no. 6, 1327–1343. MR**1878322**, 10.1007/s00039-001-8232-6**[Why04]**-,*Geometries which fiber over trees*, in preparation, 2004.