On uniform exponential growth for linear groups

By A. Eskin, Shahar Mozes, H. Oh. 2001

127 citations recorded by the source corpus.

Let Γ be a finitely generated group. Given a finite set of generators S of Γ, the word length lS(γ) for an element γ ∈ Γ is defined to be the smallest positive integer for which there exist s1, · · · , sn ∈ S ∪ S −1 such that γ = s1 · · · sn. For each n ∈ N, denote by BS(n) the set of elements in Γ whose word length with respect to S is at most n. It follows from the subadditive property of lS(·) that limn→∞ |BS(n)| 1/n exists, which we denote by ωS(Γ). A finitely generated group Γ is said to be of exponential growth if ωS(Γ) > 1, of polynomial growth if for some c > 0 and d ∈ N, |BS(n)| ≤ c · n d for all n ≥ 1 and of intermediate growth otherwise, for some finite generating set S of Γ. Observe that the growth type of Γ does not depend on the choice of generating set S. If a finitely generated group Γ is linear, it is known that Γ is either of polynomial growth in which case Γ is virtually nilpotent, or of exponential growth otherwise ([Tit72], [Mil68], [Wol68]).

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a78273f43e7152a3187b576b85cec0a22845f4a8

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Citations in the corpus

Growth and generation in SL_2(Z/pZ) (2005, background): “e argument for the sake of notational clarity. 1.5. Further remarks. There is a rich literature on the growth of sets in linear algebraic groups over ﬁelds of characteristic zero: see, most recently, [EMO]. In such a situation, one has access to topological arguments without clear analogues in Z/pZ.”
Milnor's Problem on the Growth of Groups and its Consequences (2011, background): “Oh [80]), subgroups of the mapping class group which have exponential growth (J.”
On the joint spectral radius for isometries of non-positively curved spaces and uniform growth (2018, unknown): “Then the lemma follows by “escape from subvarieties” (see [38] [21, Lemma 3.”
Groups acting on manifolds: around the Zimmer program (2008, background): “graph for Γ grows at least as λr for some λ>1 that does not depend on the choice of generators. For linear groups, exponential growth implies exponential growth by a theorem of Eskin, Mozes and Oh [50]. There are examples of groups having exponential but not uniform exponential growth due to Wils”
Recent progress on Bernoulli convolutions (2016, background): “See [19] and [14] for related earlier work.”
Growth in SL_3(Z/pZ) (2008, methodology): “SLn, n>3. There is a rich literature on growth in inﬁnite groups, based on the works of Gromov, Tits et al. There seems to be now at least one point of intersection with it: the escape argument of [EMO] will be used time and again in the course of this paper. In essence, it tells us that we may a”
Growth and generation in $\mathrm{SL}_2(\mathbb{Z}/p \mathbb{Z})$ (2008, methodology): “The following lemma4 is based closely on [ EMO ,”
A Strong Tits Alternative (2008, methodology): “In [18], Eskin-Mozes-Oh made clever use of the Cayley-Hamilton theorem in order to show a statement of a similar nature which also gives a bound on the smallest appropriate n.”