Geodesic
The geometry of subgroup embeddings and asymptotic cones
Jarnevic, Andy1
1Vanderbilt University, Nashville, TN, United States
Algebraic and Geometric Topology, Tome 25 (2025) no. 2, pp. 699-719
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Given a finitely generated subgroup H of a finitely generated group G and a nonprincipal ultrafilter ω, we consider a natural subspace, Cone ⁡ Gω(H), of the asymptotic cone of G corresponding to H. Informally, this subspace consists of the points of the asymptotic cone of G represented by elements of the ultrapower Hω. We show that the connectedness and convexity of Cone ⁡ Gω(H) detect natural properties of the embedding of H in G. We begin by defining a generalization of the distortion function and show that this function determines whether Cone ⁡ Gω(H) is connected. We then show that whether H is strongly quasiconvex in G is detected by a natural convexity property of Cone ⁡ Gω(H) in the asymptotic cone of G.

DOI : 10.2140/agt.2025.25.699
Keywords: asymptotic cones, convexity, distortion

Jarnevic, Andy  1

1 Vanderbilt University, Nashville, TN, United States 
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Jarnevic, Andy. The geometry of subgroup embeddings and asymptotic cones. Algebraic and Geometric Topology, Tome 25 (2025) no. 2, pp. 699-719. doi: 10.2140/agt.2025.25.699

[1] T Aougab, M G Durham, S J Taylor, Pulling back stability with applications to Out(Fn) and relatively hyperbolic groups, J. Lond. Math. Soc. 96 (2017) 565 | DOI

[2] J A Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space, Geom. Topol. 10 (2006) 1523 | DOI

[3] G R Conner, C Kent, Local topological properties of asymptotic cones of groups, Algebr. Geom. Topol. 14 (2014) 1413 | DOI

[4] L Van Den Dries, A J Wilkie, Gromov’s theorem on groups of polynomial growth and elementary logic, J. Algebra 89 (1984) 349 | DOI

[5] C Druţu, S Mozes, M Sapir, Divergence in lattices in semisimple Lie groups and graphs of groups, Trans. Amer. Math. Soc. 362 (2010) 2451 | DOI

[6] C Druţu, M Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005) 959 | DOI

[7] A Genevois, Hyperbolicities in CAT(0) cube complexes, Enseign. Math. 65 (2019) 33 | DOI

[8] M Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981) 53 | DOI

[9] M Gromov, Asymptotic invariants of infinite groups, from: "Geometric group theory, II", Lond. Math. Soc. Lect. Note Ser. 182, Cambridge Univ. Press (1993) 1

[10] C Kent, Asymptotic cones of HNN extensions and amalgamated products, Algebr. Geom. Topol. 14 (2014) 551 | DOI

[11] K A Mihailova, The occurrence problem for direct products of groups, Mat. Sb. 70(112) (1966) 241

[12] A Y Olshansky, On the distortion of subgroups of finitely presented groups, Mat. Sb. 188 (1997) 51 | DOI

[13] P Papasoglu, On the asymptotic cone of groups satisfying a quadratic isoperimetric inequality, J. Differential Geom. 44 (1996) 789

[14] T R Riley, Higher connectedness of asymptotic cones, Topology 42 (2003) 1289 | DOI

[15] E Rips, Subgroups of small cancellation groups, Bull. Lond. Math. Soc. 14 (1982) 45 | DOI

[16] S Thomas, B Velickovic, Asymptotic cones of finitely generated groups, Bull. Lond. Math. Soc. 32 (2000) 203 | DOI

[17] H C Tran, On strongly quasiconvex subgroups, Geom. Topol. 23 (2019) 1173 | DOI

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