Here is a list of some questions that came to my mind at some point. I found them interesting but did not have the time to fully investigate them. Feel free to send me comments or relevant references.
Is it decidable whether the fundamental group of a compact nonpositively curved cube complex is free? It may not be easy to determine whether a cubulable group is (virtually) free, or equivalently if the median graph on which it acts is a quasi-tree. The bottleneck criterion is often useful for detecting quasi-trees, but a criterion based on the combinatorics of hyperplanes would be more relevant here. For instance, it would allow one to determine when cocompact special groups are free.
Let G be a group acting geometrically on a median graph X and let g be a contracting element of G. Does there exist some n large enough such that G/<<g^n>> is cubulable? Is it special if G acts specially on X?
Construct interesting examples of groups with satisfying the fixed point property FW (i.e. every action on a median graph stabilises a cube) but not Kazhdan's property (T). When does a hyperbolic group satisfy FW?
Which finitely generated solvable groups act properly on median graphs? Polycyclic groups acting properly on median graphs are virtually abelian. This is essentially due to the fact that finitely generated abelian groups have very few proper actions on median graphs. For solvable groups, countable abelian groups have to be considered, whose actions are more delicate to analyse.
When does an infinite direct product of finite group act properly on a median graph? Some of them admit such actions but others satisfy Bergman's fixed point property.
The cubical dimension of a group is the smaller cubical dimension of a median graph on which it acts properly (and cocompacty). Find interesting lower bounds on cubical dimensions of cubulable groups.
Is the uniform membership problem for median-cocompact subgroups in cubulable groups solvable? A positive answer is obtained in arxiv:2011.05374 for convex-cocompact subgroups in torsion-free cubulable groups. But median-cocompact subgroups (i.e. subgroups acting cocompactly on connected median subgraphs) are more natural. In particular, they include all the virtually abelian subgroups.
Are median-cocompact subgroups in right-angled Artin groups separable?
A special group G always embeds in some right-angled Artin group. Let R(G) denote the smallest number of vertices of the graph defining such a right-angled Artin group. Is there a nice haracterisation of R(G)?
Is a virtually cubulable group necessarily CAT(0)? This is of course related to the famous open problem of whether virtually CAT(0) groups are CAT(0).
The known examples of virtually cubulable but not cubulable groups come from virtually abelian groups. What are other sources of such a phenomenon?
Does there exist a finitely generated group admitting a proper action on a median graph (i.e. with finite vertex-stabilisers) but not admitting a metrically proper action on a median graph?
Can we use the formalism from arxiv:1709.01258 to prove that a cocompact special group G is either relatively hyperbolic or algebraically thick? that a malnormal subgroup in G is necessarily Morse?
Introduce a reasonable definition of random cubes of finite groups. Are their fundamental groups cubulable? special?
Right-angled mock reflection groups, as studied by R. Scott, are the groups admitting generators of order two whose Cayley graphs are median. Are these groups virtually (cocompact) special? Are there interesting examples coming from a generalisation to quasi-median or mediangle graphs?
When is the outer automorphism group of a right-angled Artin group acylindrically hyperbolic? For the automorphism group, a complete answer can be found in arxiv:1807.00622.
The (outer) automorphism group of a free group does not belong to any the usual families of groups one meet in geometric group theory. Usually, this is proved using transvections. So what does happen if one avoids the transvections? What about the subgroup generated by the partial conjugations (which can be nicely described as the motion group of finitely many circles in the Euclidean space)? More generally, is the group of conjugating automorphisms of a right-angled Artin group CAT(0)? cubulable? Helly?
Describe the automorphism groups of graph products in terms of their vertex-groups. Some information can be found in arxiv:1807.00622 and references therein, but the global picture is not fully understood.
Is there a graph product of finite groups that is not quasi-isometric to a right-angled Coxeter group? A good strategy could be the consider the hyperbolic case and to use JSJ decompositions.
Given a graph G and an integer p, let C(G,p) denote the graph product over G all whose vertex-groups are cyclic of order p. Can C(G,p) and C(G,q) be quasi-isometric for p distinct from q?
Let A be a right-angled Artin group, u one its canonical generators, and g an element. Is it true that u is trivial in A/<<g>> if and only if g is conjugate to u?
Let G be an HNN extension of a right-angled Artin group conjugating two canonical generators u and v. Is it true that G is a right-angled Artin group if and only if u or v is an isolated vertex in the graph defining the Artin group?
Prove that the asymptotic cones of a graph product whose vertex-groups have simply connected (resp. contractible) asymptotic cones are also simply connected (resp. contractible). The intuition to keep in mind is that a graph product admits median structure relative to its vertex-groups. As an illustration, it is proved in arxiv1807.00622 that the asymptotic cones of a graph product embed in a product of tree-graded spaces. A more precise and more conceptual characterisation would be interesting.
In graph theory, weakly modular graphs are defined as the graphs defining two specific conditions, referred to as the triangle and quadrangle conditions. Their nonpositive curvature has been investigated in arxiv:1409.3892. However, such graphs are "made of" 3- and 4-cycles. A natural generalisation, suggested in arxiv:2212.06421, proposes to replace the quadrangle condition with a cycle condition. This new family of graphs include of course weakly modular graphs but also, for instance, subdivided one-skeletons of small cancellation complexes. Investigating further such graphs would be interesting. For instance, is there a local criterion?
Do mediangle graphs, as introduced in arxiv:2212.06421, admit a cell structure making them contractible? CAT(0)?
In metric graph theory, partial cubes refers to the graphs that isometrically embed in hypercubes. Median graphs are partial cubes. But there are many families of graphs interpolating between median graphs and partial cubes, such as semi-median and almost median graphs. Can we find interesting applications of such graphs in geometric group theory?
Let G be a one-relator quotient of a free group between some non-trivial group H and a free group of rank at least two. Is G acylindrically hyperbolic? It is at least SQ-universal according to arXiv:0603468.
Conjecture: The automorphism group of a finitely generated acylindrically hyperbolic group is acylindrically hyperbolic. The conjecture has been verified for hyperbolic groups, most relatively hyperbolic groups (including multi-ended groups), and graph products (including right-angled Artin/Coxeter groups).
If a cubical group is acylindrically hyperbolic, does it admit an acylindrical action on a hyperbolic median graph? (A possible strategy, towards a negative answer could be the following. If such an action exists, then hyperplane-stabilisers provide Morse codimension-one subgroups. In right-angled Artin groups, Morse subgroups of infinite index are necessarily free. It is not impossible that a one-ended right-angled Artin group cannot contain a free codimension-one subgroup.)
When is a (right-angled Coxeter group)-by-cyclic relatively hyperbolic?
Introduced in arxiv:1909.04318, excentric subspaces are, roughly speaking, the smallest Morse subspaces that are not hyperbolic. Interestingly, it turns out that, in graph products of finite groups (such as right-angled Coxeter groups), excentric subspaces are (up to finite Hausdorff distance) subgroups. This is useful in producing interesting quasi-isometric invariants. Can such a phenomenon be found in other families of groups?
Acylindrically hyperbolic groups provide a very large class of negatively curved groups. But, among them, is there some natural way to say that some groups are "more hyperbolic" than others? A possibility is to use divergence or thickness to quantify such a behaviour. But how does it translate algebraically?
Solve the Isomorphism problem for the symmetrisations QV_{n,m} of Thompson's groups V_{n,m}.
Has Brin-Thompson's group nV the fixed-point property FW (i.e. every action on a median graph has a cube globally stabilied)? Is the group a-T-menable? It is proved in arxiv:2208.00685 that nV contains distorted elements, so at least we know that it cannot act properly on a median graph.
Are Thompson's groups V and F quasi-isometric? Of course, the expected answer is negative. But which invariant can we use to prove it?
Can Brin-Thompson's groups nV and mV be quasi-isometric for distinct n and m?
Describe the asymptotic cones of Thompson's group F. The motivation is to show that they are not products and to conclude that F and FxZ are not quasi-isometric.
Are there reasonable Thompson-like groups of finite cohomological dimension?
Are there reasonable Thompson-like groups that are acylindrically hyperbolic? A motivation comes from the analogy between Thompson's groups and Cremona groups, the latters being are acylindrically hyperbolic. Examples can be looked for among diagram groups. For instance, the diagram group D(P,x) defined by the semigroup presentation P = < x | x=x², x=x^3 > is acylindrically hyperbolic. But is it reasonably a Thompson-like group?
Motivated by big mapping class groups, proper homotopy equivalences of infinite graphs are studided in arxiv:2109.06908. Are there groups constructed in a similar way as Thompson-like groups obtained from big mapping class groups by considering asymptotically rigid homeomorphisms? More concretely, let LT denote the infinite binary tree with a loop attached at each vertex. Consider the group of the proper homotopy equivalences LT -> LT that are planar isometries outside a bounded subset. The group surjects onto Thompson's group T with a kernel isomorphic to a direct limit of outer automorphism groups of free groups. Is the group of finite F_\infty?
Röver's group is a finitely presented group naturally generated by Thompson's group V and Grigorchuk's group. These two groups are known to act properly on median graphs, so is it also possible to make Röver's group act properly on a median graph?
Which right-angled Artin groups can be described as diagram groups? Is every right-angled Artin group a subgroup of a diagram group?
Are relatively hyperbolic diagram groups free products?
If a diagram group does not contain a subgroup isomorphic to the lamplighter group ZwrZ, does it embed in a right-angled Artin group?
Study diagram groups given by ramdomly chosen semigroup presentations. Generically, do they contain Thompson's group F? or the lamplighter group ZwrZ? are they residually finite?
Are free product V*V and the wreath product VwrZ/2Z symmetric diagram groups?
If a diagram group does not contain a subgroup isomorphic to Thompson's group F, is it residually finite?
Is it true that a diagram group cannot contain a subgroup isomorphic to the wreath product ZwrZ²?
Are there non-abelian finitely presented solvable diagram groups?
Which groups can be described as graph braid groups? For instance, are there many one-ended hyperbolic groups?
Which right-angled Artin groups can be described as graph braid groups?
When is a graph braid group free?
Fixing a number of particles, how can we decide whether two graphs have the same braid group?
When is a graph braid group relatively hyperbolic?
We know that a wreath product (finite group)wrZ^n acts properly on a median graph of cubical dimension 2n. Can we do better? i.e. does it admit a proper action on a median graph of cubical dimension 2n-1?
Is it true that every proper action of the lamplighter group ZwrZ on a CAT(0) space contains a parabolic? The question is motivated by arxiv:1209.5804, which shows that the action of Thompson's group F on its CAT(0) cube complex admits parabolic isometries. If the previous question has a positive answer, then this shows that the phenomenon is not specific to this particular action and that this is already true for the subgroup ZwrZ < F.
Which groups are quasi-isometric to wreath products of the form (finite group)wrZ²?
Which big mapping class groups are SQ-univeral? It is known that some big mapping class groups admit actions on hyperbolic spaces with WWPD elements. So it is natural to ask: if a group acts on some hyperbolic space with independent WWDP elements, is it necessarily SQ-universal?
In horospherical products of at least three regular trees, are there uniform lattices that are not virtually solvable? The question is motivated by another well-known question: Is the property of being virtually solvable preserved by quasi-isometries among finitely presented groups?
Give an example of an infinitely presented group such that only countably many groups are quasi-isometric to it.
What are the Dehn functions of Houghton's groups?
Describe the possible finite subgroups in cactus groups. See arxiv:2212.03494 and references therein for more details.
Let G be a (locally finite)-by-Z^n group. If G is of type F_{n+1}, is it necessarily virtually Z^n?
Is it true that a 2-generated subgroup in Higman's group is isomorphic to {1}, Z, F_2, or a subgroup of BS(1,2)?