# Are all almost virtually free groups word hyperbolic?

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 $$F_n$$.

Answered by user29123 on January 3, 2022

## Related Questions

### Invariant $SU(3)$ subgroup for ${bf 8}$ in ${bf 3}^* otimes {bf 3} ={bf 1} oplus {bf 8}$

1  Asked on December 3, 2021 by annie-marie-cur

### Proof that a continuous function with continuous right derivatives is differentiable.

1  Asked on December 3, 2021

### Finding the volume when a parabola is rotated about the line $y = 4$.

1  Asked on December 3, 2021

### Differential equation, modulus signs in solution?

2  Asked on December 3, 2021 by refnom95

### Consider the sequence where $a_1>0$, $ka_n>a_{n+1}$ and $0<k<1$. Can we say it converges?

1  Asked on December 3, 2021 by oek-cafu

### In a Reflexive banach space, given a closed convex set $C$ and some point $y$, there is a point in $C$, of minimal distance to $y$

2  Asked on December 3, 2021

### Ball / Urn question with a twist

1  Asked on December 3, 2021 by user109387

### How to prove $phi'(t)1_{Omega_t}(w)$ is measurable?

1  Asked on December 3, 2021 by czzzzzzz

### Using characteristic functions to determine distribution of sum of independent normal random variables.

0  Asked on December 3, 2021 by jkeg

### Radioactivity formula using differential equations?

3  Asked on December 3, 2021 by mitali-mittal

### What is the algebraic interpretation of a contracted product?

0  Asked on December 3, 2021 by james-steele

### $f:[0,1]rightarrow[0,1]$, measurable, and $int_{[0,1]}f(x)dx=yimplies m{x:f(x)>frac{y}{2}}geqfrac{y}{2}$.

1  Asked on December 3, 2021

### Evaluating an integral with a division by $0$ issue

3  Asked on December 1, 2021

### Evaluating $sumlimits_{i=lceil frac{n}{2}rceil}^inftybinom{2i}{n}frac{1}{2^i}$

1  Asked on December 1, 2021 by maxim-enis

### Lie group theory’s connection to fractional calculus?

0  Asked on December 1, 2021

### Solution to autonomous differential equation with locally lipscitz function

1  Asked on December 1, 2021

### Let $A,Bin M_n (mathbb R), lambdain sigma(B), alpha in sigma (B)$ and  be inner product show $=lambda||x||^2+alpha$

0  Asked on December 1, 2021

### Is an automorphism a function or a group?

2  Asked on December 1, 2021 by josh-charleston

### Fundamental Theorem of Projective Geometry for Finite Fields

0  Asked on December 1, 2021 by am2000

### Prove a metric space is totally bounded

2  Asked on December 1, 2021