# Is residual finiteness a property of “many” finitely presented groups?

Is there a reasonable random model for selecting a finitely presented group $G$ such that with positive probablity (or even with probability almost $1$) some of the following properties hold:

1. $G$ is residually finite.
2. $G$ is subgroup seperable (LERF).
3. The first $l_2$ Betti number of $G$ is positive.
4. The cost of $G$ is greater than 1.
• Dear @StefanKohl I know that there exists a notion of generic finitely presented groups studied by (say) Arzhantseva and Olshanskii. They consider the proportion of finitely presented groups satisfying a certain property by fixing the number of generators and relations and go over all groups with these relations having length at most $M$. Then you take $M \to \infty$ and see what is the limiting probability. There probably exist other random models. I have heard that Gromov proved (or conjectured) that most groups are hyperbolic. In my question I ask if random groups are residually finite. – Pablo Jan 2 '15 at 15:02
• @StefanKohl for generic properties of groups see ams.org/journals/proc/2000-128-11/S0002-9939-00-05508-8 or researchgate.net/publication/… so I want to know if residual finiteness is generic in this sense (or in a slightly different one). For results on one relator groups see Theorem 1.6 in arxiv.org/pdf/1001.2829.pdf. For different definitions of a "random model" one can consult yann-ollivier.org/rech/publs/randomgroups.pdf – Pablo Jan 2 '15 at 15:06
• I believe you are correct in saying that there are widely accepted definitions of random finitely presented groups, and significant results (by Gromov?) such as almost all finitely presented groups are hyperbolic. But I have never encountered a definition of a random finitely generated group, so perhaps you should frame your question for finitely presented groups. – Derek Holt Jan 2 '15 at 15:19
• @DerekHolt I am ready to restrict my attention to finitely presented groups (edited the question). – Pablo Jan 2 '15 at 15:41
• @Derek: about f.g. groups rather than f.p.: there is a space of marked f.g. groups on $k$ generators which is a nice compact space. In this space there are topological notions of large subsets, e.g. $G_\delta$ subsets, or the intersection of $G_\delta$ subsets with subsets of countable complement. Perhaps there are natural classes of measure but I don't know. – YCor Jan 2 '15 at 16:39

A random group at density less than 1/6 is known to be the fundamental group of a compact, non-positively curved cube complex, by Ollivier--Wise. By Agol's theorem, all such hyperbolic groups are virtually special, and hence residually finite, QCERF, etc.

• but is it also LERF? This was my second question. – Pablo Jan 4 '15 at 12:10
• Pablo, LERF isn't really the right notion here. One can ask whether every fg subgroup is qc. This is known to fail for random 2-generator 1-relator groups, and presumably it fails in general. – HJRW Jan 4 '15 at 14:00
• Thanks for this clarification. And one last thing on this issue, what about my third question? Do random groups at density $< 1/6$ have positive first $l_2$ Betti number? In fact I am really interested in the positivity of the rank gradient. – Pablo Jan 4 '15 at 14:28
• Pablo, I think this is an interesting question. The result I mentioned about 2-generator 1-relator groups is in a paper of Dunfield and (Dylan) Thurston. They prove that a random 2-generator 1-relator group is fg-free-by-cyclic with probability strictly between 0 and 1. In particular, at least some of the time, such groups have zero rank gradient. As far as I know, nothing is known about models with more generators or relations. – HJRW Jan 4 '15 at 14:54
• (But you might like to check out the work of Sapir and his co-authors on random 1-relator groups with more generators.) – HJRW Jan 4 '15 at 14:55

According to Henry Wilton (who will probably comment extensively himself), the question of whether a random finitely presented group is residually finite was wide open in 2012 (and I am pretty sure not enough has changed now, outside of 3-manifold group context).

• Yes it has changed a lot, because several random models (e.g. Gromov's model at low density) provides almost surely $C'(1/6)$ f.p. groups, which are residually finite by Agol's result (combined with the Haglund-Wise theory). This goes far beyond 3-manifold groups. – YCor Jan 2 '15 at 16:30
• @YCor Yves knows a lot more, so maybe he should post an answer :) – Igor Rivin Jan 2 '15 at 16:41
• @YCor is there a chance that in this model we also have subgroup separability (LERF) almost surely? or with positive probability? – Pablo Jan 2 '15 at 23:38
• I think it's worth noting that my blog post appeared the week before Agol's preprint proving the virtual Haken conjecture (as explained in the subsequent blog post). As Yves says, this changed everything. – HJRW Jan 4 '15 at 11:49