## Some questions and a few answers |

Here is a selection of some questions for which I have contributed answers.

Below this are a few questions that I asked; I include those answers that I am aware of.

**Disclaimer:** This page is, at best, a cabinet of curiosities. I don't view mathematics as an intellectual safari; much of my work and joint work on cohomology, on CAT(0) geometry and on groups is not directed at "problem-solving".

Below this are a few questions that I asked; I include those answers that I am aware of.

- In 1929, J. von Neumann highlighted the fact that the Hausdorff–Banach–Tarski paradox relies fundamentally on the presence of a (non-abelian) free subgroup in the symmetry group under consideration. The question whether a group without such free subgroups could lead to paradoxical decompositions later became famous as the von Neumann Problem, a label probably due to M. Day.

The first counter-examples were found by A. Olshanskii and Adyan–Novikov in the early 1980s. These remarkably difficult examples were later followed by several other constructions.

I gave here a concrete family of simple examples. For instance, the group of all piecewise projective homeomorphisms of the real line is paradoxical and without free subgroups. - In 1950, J. Dixmier asked whether all unitarisable groups are amenable. Here a group is called unitarisable if every representation on a Hilbert space can be conjugated to a unitary representation, provided it is uniformly bounded (an obvious necessary condition).

This question is open and until recently the only known non-unitarisable groups contained free subgroups. In fact G. Pisier asked specifically whether Burnside groups could be unitarisable.

With I. Epstein, we established here a criterion which implies for instance that some torsion groups are non-unitarisable. Then we proved here with N. Ozawa that in fact every non-amenable group becomes non-unitarisable after forming the wreath product with any infinite abelian group. In particular, it follows that many Burnside groups are non-unitarisable. With M. Gheysens, we extended the latter work to locally compact groups here. - In 1957, M. Day asked if the class of amenable groups introduced by J. von Neumann in 1929 contains more than the elementary amenable groups, i.e. the groups obtained from finite and abelian groups by the operations preserving amenability already described by J. von Neumann.

The first non-elementary example was provided in 1984 by R. Grigorchuk and more have been found since then.

Perhaps the most definitive way to exclude elementarity would be to have a simple finitely generated infinite amenable group. With K. Juschenko, we proved the existence of such groups here. - The 1960s Derivation Problem, due to J.H. Williamson, is to show that every derivation of every group algebra is inner (in the generality of locally compact groups; the algebra is in the L
^{1}sense). It was finally solved by V. Losert here in 2008 (though not with the optimal constant).

With U. Bader and T. Gelander, we proved here a general fixed point theorem for L^{1}-spaces. As a special case, it gives an elementary proof of the Derivation Problem and establishes the optimal constant. - In 1968, H. Reiter asked if the amenability of a closed subgroup H of a locally compact group G is characterised by the existence of (right, bounded) approximate units in the algebra J
^{1}(G,H). This algebra is defined as the kernel of the integration map L^{1}(G)→L^{1}(G/H).

With P.-E. Caprace, we observed here that this is indeed true. - In the 1970s, J. Wiegold asked if a finitely generated perfect group can always be normally generated by a single element. In other words, can the group be "killed" by adding a single relation?

Although this is the case for finite groups, it is believed to be false in general. The question makes sense for locally compact groups too, but the conjectured counter-examples for discrete groups would a fortiori provide counter-examples for any locally compact group admitting them as a quotient.

Perhaps surprisingly, we proved here with A. Eisenmann that Wiegold's question has a positive answer for the locally compact groups that do not admit infinite discrete quotients. Our proof relies heavily on this joint work with P.-E. Caprace. - In his 1972 monograph, P. Eymard proposed the following problem. A subgroup H<G is called co-amenable if it has a relative fixed point property, or equivalently if there is a G-invariant mean on G/H. It is easy to see that this property is transitive when considering three nested groups K<H<G, and that the co-amenability of K<G implies that of H<G. Eymard's problem was whether it implies the co-amenability of K<H.

If all subgroups are normal, this amounts simply to the amenability of the quotient group and hence Eymard's problem reduces to the well-known fact that subgroups of amenable groups are amenable.

However, with S. Popa, we gave here a simple family of counter-examples, including situations with K normal in H and simultaneously H normal in G. See also this work by V. Pestov. - The Zimmer Program initiated in the 1980s by R. Zimmer aims notably at classifying the actions of lattices in simple Lie groups on manifolds of small dimension. This would constitute a non-linear extension of G. Margulis' super-rigidity. Notably, it is conjectured that such actions are essentially trivial if the dimension of the manifold is strictly below the rank of the Lie group.

The lowest dimensional case regards actions on the line and the circle; even this remains open in full generality. With M. Burger, we established here and here cohomological vanishing results which imply this conjecture in the differentiable case and yields a partial answer in the continuous case. Another proof was found at the same time by É. Ghys here. - A. Katok asked whether there exists (non-abelian) free groups of interval exchange transformations (IET).

If the rank of the translation part of an IET group does not exceed two, then we proved here with K. Juschenko, N. Matte Bon and M. de la Salle that it cannot contain such free subgroups. (We proved the stronger statement that such low rank IET groups are amenable.) - G. Margulis conjectured that higher rank simple algebraic groups have the fixed point property for isometric actions on uniformly convex Banach spaces. This would be a considerable strengthening of D. Kazhdan's property (T).

With U. Bader, A. Furman and T. Gelander, we proved here that this conjecture holds in the (very special) case of L^{p}-spaces.

Much stronger partial solutions have since then been established by V. Lafforgue, T. de Laat and M. de la Salle. - D. Gaboriau asked if a closed invariant subspace E of ℓ
^{p}(G) is necessarily trivial when it meets trivially an increasing exhaustion of ℓ^{p}(G) by closed invariant subspaces. This is known to hold for p=2.

With H. Petersen, we proved here that this fails for all p>2 provided G is not torsion, or more generally contains an infinite elementary amenable subgroup.

In other words, submodules of ℓ^{p}(G) do not constitute a "continuous geometry".

- Let v
_{p}:**Q*** →**Z**be the p-adic valuation and define D_{p,q}(x) = v_{p}(x) v_{q}(1−x) − v_{q}(x) v_{p}(1−x) for x≠0,1. Is every (non-trivial, possibly infinite) linear combination of D_{p,q}over all primes p,q unbounded?

The reference and background for this question is here, Question 1.3 (or here, Problem R). - Prove that the group SL
_{2}(**Z**√2) does not admit any faithful transitive amenable action on any set.

The reference and background for this question is here, Section 4.I (with Y. Glasner). - Consider a bounded complete CAT(0) space (for instance, a bounded closed convex set in your favourite complete CAT(0) space). Does it necessarily admit an extremal point?

(Question proposed in a 2006 Master Class at CIRM.) Update: a counter-example is provided here! - Prove that the (continuous) cohomology of a connected semi-simple Lie group with finite centre coincides with its bounded cohomology.

A reference for this question is here, Problem A; earlier, J. Dupont asked whether, at least, all characteristic classes are bounded.

With M. Burger, we solved Problem A in degree 2 and, for the special case of SL_{2}(**R**), also in degree 3. In this special case, the degree 4 has meanwhile been established here by T. Hartnick and A. Ott. - Consider the group SL
_{3}(**Z**). Is it true that every unitary ε-representation is δ-close to an actual representation, where ε>0 depends only on δ>0?

The reference for this question is here, Problem F'. A negative answer has been established here by M. Burger, N. Ozawa and A. Thom, but they give a positive answer provided only finite-dimensional representations are considered. - Consider a non-amenable group with an action on a measure space X that is amenable in Zimmer's sense. For which integers n can the diagonal action on X
^{n}be ergodic? Can we have n≥4?

The reference and background for this question is here, Problems G, H and I. - Is there a geometric characterisation of the non-vanishing of the second bounded cohomology with coefficients in the regular representation of a group? (This property seems to be related to negative curvature.) In particular, is it a quasi-isometry invariant for finitely generated groups?

The reference and background for this question is here, Problem J. - Consider a non-amenable group. The existence of paradoxical decompositions can be rephrased as the existence of certain (non-elementary) regular spanning forests on the group. Does the space of such forests admit an invariant probability measure?

The reference for this question is here, Problems K and K'. A positive answer has been announced by G. Kun here. - Let k be a countable field of infinite transcendence degree over its prime field and let n≥3. Does the group SL
_{n}(k[X]) have non-trivial quasi-morphisms?

The reference for this question suggested by M. Abért is here, Problem Q. When k is Euclidean and n≥6, M. Mimura has proved here that there are no non-trivial quasi-morphisms. - Prove that every unitarisable group has trivial cost.

Or prove the a priori weaker statement that its first L^{2}-Betti number vanishes.

A partial result is here (with I. Epstein); for instance, we prove these statements for residually finite groups. - Let G be a non-amenable group. Is there an integer n such that G
^{n}is not unitarisable?

Is, at least, the restricted product G^{(∞)}non-unitarisable? - Consider a finitely generated non-unital ring R and suppose that it is idempotent, i.e. R·R=R. Find an example that is not generated by less than 10 elements as an ideal in itself.

The reference and background for this question is here (with N. Ozawa and A. Thom). - Classify the representations of the isometry group of the n-dimensional hyperbolic space into the infinite-dimensional Lie group O(p,∞).

The reference and background for this question is here, Chapter 5 (with P. Py).

For instance, we prove that 2<p<n is excluded for irreducible representations when n>4. - Prove that the submodules of ℓ
^{p}(G) do not constitute a "continuous geometry" when p>2.

The reference and background for this question is here, Conjecture 3 (with H. Petersen).

For instance, we proved this if G is not torsion, or more generally contains an infinite elementary amenable subgroup. - Consider the space of closed subgroups of a given locally compact group, endowed with the Chabauty topology. Is the set of amenable subgroups closed?

The reference and background for this question is here (with P.-E. Caprace). - Consider the action of SL
_{3}(**R**) on the projective plane. Prove that there is no equivariant linear lifting for bounded function classes.

The reference and background for this question is here, Problem 10.

For instance, there does exist a lifting for the analogous question about SL_{2}(**R**) and the projective line. - Let G be a countable group (or more generally a locally compact second countable group). Suppose that G has a fixed point for every continuous affine action on a Hilbert space that preserves a compact set. Does it follow that G is amenable?

The reference and background for this question is here, Question 32 (with M. Gheysens).

For instance, we prove that it follows that G does not contain non-abelian free subgroups. We also answer the question if compactness is replaced by weak compactness. - Can a Kazhdan group act on a dendrite without a fixed point?

This question breaks naturally into two sub-questions of a rather different nature:

(i) Can a Kazhdan group be left-orderable?

(ii) Is there a non-elementary action of a Kazhdan group on a dendrite?

Question (ii) is asked in this joint paper with B. Duchesne. An action is called elementary if it fixed a point or preserves an arc. This turns out to be a natural notion of "triviality" for group actions on dendrites (loc. cit.). Now if a group preserves an arc, then (upon possibly passing to an index two subgroup) we are reduced to an orderability question by a well-known procedure. - Is the flatmate complex of a Euclidean building acyclic? With M. Bucher, we ask this question here.

More precisely, the flatmate complex refers to the subcomplex of the abstract chain complex defined by considering only flatmate simplices, that is, tuples of points that are contained in a same flat. We conjecture that this complex is not only acyclic, but admits a conrtacting homotopy that is bounded for the L^{1}-norm (in each dimension). We prove that this would imply the triviality of the bounded cohomology of algebraic groups over non-Archimedean local fields.

[Publications][Nicolas Monod]