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Authors
Eric Jaligot
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We build two non-abelian CSA-groups in which maximal abelian subgroups are conjugate and divisible.
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We introduce an obstruction to the existence of a coarse embedding of a given group or space into a hyperbolic group, or more generally into a hyperbolic graph of bounded degree. The condition we consider is admitting exponentially many fat bigons, and it is preserved by a coarse embedding between graphs with bounded degree. Groups with exponential growth and linear divergence (such as direct products of two groups one of which has exponential growth, solvable groups that are not virtually nilpotent, and uniform higher-rank lattices) have this property and hyperbolic graphs do not, so the former cannot be coarsely embedded into the latter. Other examples include certain lacunary hyperbolic and certain small cancellation groups.
We consider finite sums of counting functions on the free group $F_n$ and the free monoid $M_n$ for $n geq 2$. Two such sums are considered equivalent if they differ by a bounded function. We find the complete set of linear relations between equivalence classes of sums of counting functions and apply this result to construct an explicit basis for the vector space of such equivalence classes. Moreover, we provide a graphical algorithm to determine whether two given sums of counting functions are equivalent. In particular, this yields an algorithm to decide whether two sums of Brooks quasimorphisms on $F_n$ represent the same class in bounded cohomology.
Let $F_n$ be a free group of finite rank $n geq 2$. We prove that if $H$ is a subgroup of $F_n$ with $textrm{rk}(H)=2$ and $R$ is a retract of $F_n$, then $H cap R$ is a retract of $H$. However, for every $m geq 3$ and every $1 leq k leq n-1$, there exist a subgroup $H$ of $F_n$ of rank $m$ and a retract $R$ of $F_n$ of rank $k$ such that $H cap R$ is not a retract of $H$. This gives a complete answer to a question of Bergman. Furthermore, we provide positive evidence for the inertia conjecture of Dicks and Ventura. More precisely, we prove that $textrm{rk}(H cap textrm{Fix}(S)) leq textrm{rk}(H)$ for every family $S$ of endomorphisms of $F_n$ and every subgroup $H$ of $F_n$ with $textrm{rk}(H) leq 3$.
We give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1-ended, hyperbolic groups, namely right-angled Coxeter groups defined by generalized theta-graphs and cycles of generalized theta-graphs, and geometric amalgams of free groups whose JSJ graphs are trees of diameter at most 4. We also show that if a geometric amalgam of free groups has JSJ graph a tree, then it is commensurable to a right-angled Coxeter group, and give an example of a geometric amalgam of free groups which is not quasi-isometric (hence not commensurable) to any group which is finitely generated by torsion elements. Our proofs involve a new geometric realization of the right-angled Coxeter groups we consider, such that covers corresponding to torsion-free, finite-index subgroups are surface amalgams.
A $k$-free like group is a $k$-generated group $G$ with a sequence of $k$-element generating sets $Z_n$ such that the girth of $G$ relative to $Z_n$ is unbounded and the Cheeger constant of $G$ relative to $Z_n$ is bounded away from 0. By a recent result of Benjamini-Nachmias-Peres, this implies that the critical bond percolation probability of the Cayley graph of $G$ relative to $Z_n$ tends to $1/(2k-1)$ as $nto infty$. Answering a question of Benjamini, we construct many non-free groups that are $k$-free like for all sufficiently large $k$.
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