## Notes of Goulnara Arzhantseva’s lecture

Infinitely presented groups satisfying Baum-Connes conjectures

I am looking for counter-examples, but on the way, I expect to prove positive results. The kind of groups I have in mind are Tarski monsters, Burnside groups, Gromov monsters. I study analytic properties for such groups, like Rapid Decay and Haagerup property.

Monsters arise as direct limits of hyperbolic groups. Hyperbolic groups satisfy the strongest form of BC conjecture, with coefficients (Lafforgue 2012). This strong form also follows from Haagerup property (Higson-Kasparov 1997). On the other hand, RD merely implies the standard form of BC conjecture (Lafforgue 2002).

1. Gromov’s monster group

Data: an expanding family of graphs such that diam/girth stays bounded, a labelling of its union ${\theta}$ by a finite set ${S}$ which satisfies geometric small cancellation (a random labelling does). Then ${G_\infty =}$ where ${r_i}$ are the labels of cycles in ${\theta}$.

Theorem 1 (Higson-Lafforgue-Skandalis) ${G_\infty}$ does not satisfy the strong Baum-Connes conjecture.

Question. Where is the frontier between BC and non BC among limits of hyperbolic groups ? I expect it to be expressible in terms of small cancellation.

1.1. Small cancellation

Small cancellation means that piece is small with respect to relator containing it. Each situation specifies the words piece, small, relator.

Classical setting: finite presentation, a piece is a common subword between two (cyclic conjugates of) relators.

Graphical setting: relators are generated as labellings of cycles. Ppieces are arcs of cycles having the same labelling.

Geometric setting: involves an action on some space.

In all cases, small cancellation implies hyperbolicity.

2. Rapid Decay

Definition 2 (Haagerup 1978 for free groups, Jolissaint in general) Let group algebra ${\mathbb{C}G}$ act on ${L^2(G)}$. Say ${G}$ has property RD if there is a polynomial ${P}$ such that for all ${R>0}$, for all ${f }$ in ${\mathbb{C}G}$ vanishing outside the ${R}$-ball, for all ${g\in L^2 (G)}$,

$\displaystyle \begin{array}{rcl} |f*g|

Example 1 Groups of polynomial growth have RD.

Hyperbolic groups have RD.

Having amenable subgroups with exponential growth is an obstruction to RD (Jolissaint).

Example 2 (Thomas-Velickovic) ${G=}$, ${I}$ a subset of integers.

Theorem 3 (Arzhantseva-Drutu) Infinitely presented classical ${C'(1/24)}$-small cancellation groups have rapid decay.

The proof is quite technical (22 subcases…) but gives a fairly precise description of geodesics in such groups.

Note that lamplighter group is a direct limit of hyperbolic groups, but it is amenable of exponential growth, so no RD.

Conjecture. Rapid decay + lacunary direct limit of hyperbolic groups ${\Rightarrow}$ Baum-Connes.

3. Haagerup property

Definition 4 ${G}$ has Haagerup property if it has a proper affine sometric action on some Hilbert space.

Example 3 Amenable groups have it. (relative) property (T) is an obstruction to Haagerup property.

Fact (Haglund-Paulin-Valette): Proper action on a space with walls ${\Rightarrow}$ Haagerup.

Inspired by this fact, we find a sufficient condition for a Gromov-like class of examples to have Haagerup property.

Theorem 5 (Arzhantseva-Osajda) Let ${G_\infty = < S | r_i >}$ where ${r_i}$ are labels of cycles of graph ${\theta}$. Assume

1. ${\theta}$ has a wall structure.

2. Labelling satisfies a ${Gr'(\lambda)}$ condition.

3. Wall structure and labelling satisfy a ${\beta}$-separation property.

Then ${G_\infty}$ has Haagerup property.

Definition 6 (inspired by Wise) Let ${\beta>0}$. Let ${\Omega}$ be a finite graph with a wall structure. It satisfies ${\beta}$-separation if for every wall ${W}$, for every pair of edges ${e, e'}$ in ${W}$, if ${r}$ is the length of the largest simple cycle in ${W}$ containing ${e}$ and ${e'}$, then ${d(e,e') > \beta r}$.

Corollary 7 Infinitely presented classical small cancellation groups have Haagerup property.

Potential further corollaries (they depend on existence of labelling satisfying condition 2 in the theorem, which I cannot establish yet).

1. The exist groups which are coarsely embeddable into Hilbert spaces but do not have property A.

2. Haagerup does not imply property A.

3. Infinitely presented small cancellation groups do not have finite asymptotic dimension.