Uniform Uniform Exponential Growth of Subgroups of the Mapping Class Group

By Johanna Mangahas. 2008

75 citations recorded by the source corpus.

Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any such subgroup explicit free group generators which are “short” in any word metric. Besides bounding growth, this allows a bound on the return probability of simple random walks.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6c2b8a3f29d18baaa214ccffbb7d7631bfaa9a72

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Citations in the corpus

The geometry of right angled Artin subgroups of mapping class groups (2010, background): “The starting point is the “Behrstock inequality” [2] (see also [32], Lemma 2.”
Milnor's Problem on the Growth of Groups and its Consequences (2011, result): “]), one-relator groups, solvable groups, linear groups over ﬁelds of characteristic 0 (A.Eskin, S.Mozes and H.Oh [72]), subgroups of the mapping class group which have exponential growth (J. Mangahas [148]), and some other groups and classes of groups. The fact that solvable groups of exponential gr”
Coarse median spaces and groups (2013, unknown): “This is an immediate consequence of the result in [Behrstock 2006]; see also [Mangahas 2010].”
Stable commutator length on mapping class groups (2013, background): “On the plus side, subsurface projection constants are uniform (see Leininger’s proof in [29, 30]) and so is the hyperbolicity constant δ of curve graphs, see [10, 25, 18].”
Hierarchically hyperbolic spaces II: Combination theorems and the distance formula (2015, unknown): “coarsely surjective). Both parts of axiom 4 can be found in [Beh06]. The nesting part is elementary, and a short elementary proof in the overlapping case was obtained by Leininger and can be found in [Man10]. Axiom 7 was proven in [MM00], and an elementary proof is available in [Web15]. In fact, in ”