Rank gradient versus stable integral simplicial volume
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Rank gradient versus stable integral simplicial volume Clara Löh1
© Akadémiai Kiadó, Budapest, Hungary 2017
Abstract We observe that stable integral simplicial volume of closed manifolds gives an upper bound for the rank gradient of the corresponding fundamental groups. Keywords Stable integral simplicial volume · Rank gradient Mathematics Subject Classification 57R19 · 20E18 · 20F65
1 Introduction The residually finite view on groups or spaces aims at understanding groups and spaces through gradient invariants: If I is an invariant of groups, then we define the associated gradient invariant I for groups by I () :=
I (H ) , H ∈F() [ : H ] inf
where F() denotes the set of all finite index subgroups of . For example, the rank gradient is the gradient invariant associated with the minimal number of generators of groups (Sect. 2), originally introduced by Lackenby [12]. Further well-studied examples are the Betti number gradient and the logarithmic torsion homology gradient. Stable integral simplicial volume is the gradient invariant associated with integral simplicial volume (Sect. 3). It is known that stable integral simplicial volume yields upper bounds for Betti number gradients and logarithmic torsion homology gradients [4, Theorem 1.6, Theorem 2.6].
This work was supported by the CRC 1085 Higher Invariants (Universität Regensburg, funded by the DFG).
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Clara Löh [email protected] http://www.mathematik.uni-r.de/loeh Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
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C. Löh
In this note, we observe that stable integral simplcial volume also gives an upper bound for the rank gradient of the corresponding fundamental groups: Theorem 1.1 Let M be an oriented closed connected manifold with fundamental group and let ∗ = (k )k∈N be a chain of finite index subgroups of . Then rg(, ∗ ) ≤ MZ∗ . In particular, rg ≤ M∞ Z . The result even holds without any asymptotics (Lemma 4.2), but the gradient invariants seem to be the relevant invariants. In particular, vanishing results for stable integral simplicial volume imply corresponding vanishing results for the rank gradient. For example, we obtain an alternative argument for the following rather special case of a result by Lackenby [12, Theorem 1.2]: Corollary 1.2 Let be a residually finite infinite amenable group that admits an oriented closed connected manifold as model of the classifying space K (, 1). Then rg = 0. Proof Let M be such a model of K (, 1). Then M∞ Z = 0 [4, Theorem 1.10]. Hence, Theorem 1.1 gives rg = 0. However, the bound in Theorem 1.1 in general is far from being sharp. For instance, if is the fundamental group of an oriented closed connected surface M of genus g ∈ N≥1 , then (2) it is well known that rg = b1 = 2 · g − 2 [1][5, Proposition VI.9], but M∞ Z = M = 4 · g − 4 [8,13]. In more positive words, Theorem 1.1 shows that stable integral simplicial volume is a geometric refinement of the rank gradient. In contrast to the residual point of view, the dynam
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