Ordering the space of finitely generated groups

Abstract

We consider the oriented graph whose vertices are isomorphism classes of finitely generated groups, with an edge from G to H if, for some generating set T in H and some sequence of generating sets S_i in G, the marked balls of radius i in (G,S_i) and in (H,T) coincide. Given a nilpotent group G, we characterize its connected component in this graph: if that connected component contains at least one torsion-free group, then it consists of those groups which generate the same variety of groups as G. The arrows in the graph define a preorder on the set of isomorphism classes of finitely generated groups. We show that a partial order can be imbedded in this preorder if and only if it is realizable by subsets of a countable set under inclusion. We show that every countable group imbeds in a group of non-uniform exponential growth. In particular, there exist groups of non-uniform exponential growth that are not residually of subexponential growth and do not admit a uniform imbedding into Hilbert space

Transitivity degrees of countable groups and acylindrical hyperbolicity

“…Remark 5.5. The proof of Proposition 5.4(d) actually shows that for all k, G has a generating set with n elements and no simple cycles of length ≤ k, where n is independent of k. This is equivalent to G preforming the free group F n in the language of [6] and to G being without almost-identities in the language of [47]. However, it is unclear if this is actually stronger then the property of infinite girth (see [6,Question 8.5]).…”

“…, a n of G * x , so we obtain that G * x has finite girth. However this is well-known to be false (for example, see [6,Example 2.15]).…”

Group approximation in Cayley topology and coarse geometry Part I: Coarse embeddings of amenable groups

“…Therefore, by the aforementioned result in [BE15], the group H 3 ≀ Z, which is amenable, admits a sequence of markings (S m ) m with respect to which the marked groups converge to some (non-abelian) free marked group F k . By Theorem A, the disjoint union m Cay(H 3 ≀ Z; S m ) does not have property A.…”

Amenability versus non‐exactness of dense subgroups of a compact group

“…It implies that it is sometimes much easier to construct an LEF group with specified property than to obtain such an RF group; compare with Remark . The LEF property for finitely generated groups is closed under taking standard (restricted) wreath products ; see Lemma . This provides us room that suffices to apply variants of Hall's embedding argument ([, 1.5]) and of the absorption trick ([, Lemma 6.13]). The former is employed to reduce the number of generators to 2; the latter is used to construct two system of markings of a fixed sequence of finite groups that have considerably different behaviors at the Cayley limits (see Step 3 in the outlined proof of Theorem below).…”

“…This trick was at least observed in a work of Bartholdi and Erschler. We exhibit a form of the absorption trick, which is easily deduced from [, Lemma 6.13]. Lemma Let .…”