Mathematics Asked by Chain Markov on January 3, 2022

Suppose $G$ is a finitely generated group with a finite symmetric generating set $A$. Lets define Cayley ball $B_A^n := (A cup {e})^n$ as the set of all elements with Cayley length (in respect to $A$) $n$ or less.

Suppose $R_1, … , R_k$ are $k$ random elements chosen uniformly from $B_A^n$. Then we can define a random $k$-generated subgroup of $G$ as $H(G, A, k, n) = langle {R_1, … , R_k} rangle$.

Now, suppose, $mathfrak{X}$ is some group property closed under finitely-generated subgroups. We say, that a finitely generated group $G := langle A rangle$ is almost $mathfrak{X}$ iff $forall k in mathbb{N} lim_{n to infty} P(H(G, A, k, n)) = 1$.

The following facts are not hard to see:

The definition does not depend on the choice of $A$

The property of being almost $mathfrak{X}$ is closed under finitely-generated subgroups

A group is almost almost $mathfrak{X}$ iff it is almost $mathfrak{X}$

Moreover, a following fact was proved by Gilman, Miasnikov and Osin in «Exponentially generic subsets of groups»:

Any word hyperbolic group is either almost free or virtually cyclic

An easy corollary of this statement is:

All word hyperbolic groups are almost virtually free

My question is whether the converse is also true:

Are all almost virtually free groups word hyperbolic?

The answer is no. The paper *Generic free subgroups and statistical hyperbolicity*, by Suzhen Han and Wen-yuan Yang, proves almost virtually free for a class of groups which includes relatively hyperbolic groups.

To make sure we are on the same page I will state the result precisely in the case of relatively hyperbolic groups. Define $U^{(k)}:={(u_1,...,u_k) mid u_i in U}$. Let $G$ be a relatively hyperbolic group generated by a finite set $S$ and let $B_n$ be the ball of radius $n$ in the Cayley graph of $(G,S)$ centered at the identity. They show

$$lim_{n to infty} frac{ left|X cap B_n^{(k)}right|}{|B_n^{(k)}|} = 1$$

where $X subseteq G^{(k)}$ is the set of elements $(g_1,...,g_k)$ such that $langle g_1,...,g_k rangle $ is a free group of rank $k$ (Corollary of Corollary 1.6). In particular:

- Almost virtually free does not imply hyperbolicity since relatively hyperbolic does not imply hyperbolic (see next bullet point for an example).
- Almost virtually free groups can have subgroups which are not almost virtually free. Note that $mathbb{Z}^2$ is not almost virtually free but can be contained in relatively hyperbolic groups. If $M$ is a finite volume hyperbolic three manifold with cusps then $pi_1(M)$ is relatively hyperbolic and contains $mathbb{Z}^2$ subgroups.

I would like to point out that what is shown in *Exponentially generic subsets of groups* is somewhat different from the result above for hyperbolic groups. Essentially what they prove is that when you look at surjective homomorphism $F(S) to G$, $G$ hyperbolic, that tuples of words generically map to tuples of elements which generate a free group. This is somewhat different from the ball model of randomness and I don't believe it follows that you get the almost virtually free property for hyperbolic groups.

If instead you use this model of randomness then your question still has a negative answer. The authors of this paper point out groups which have surjective homomorphisms to non-elementary hyperbolic groups have the "*word almost virtually free property*". For example you get that $F_n times mathbb Z$ has this property, witnessed by the projection to $F_n$.

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