arXiv subjects

Alexis Marchand

Publications and source records attributed to Alexis Marchand.

4 recordsLinked to original sources

arXiv2024

From letter-quasimorphisms to angle structures and spectral gaps for scl

We give a new geometric proof of a theorem of Heuer showing that, in the presence of letter-quasimorphisms (which are analogues of real-valued quasimorphisms with image in free groups), and in particular in RAAGs, there is a sharp lower bound of 1/2 for stable commutator length. Our approach is to show that letter-quasimorphisms give rise to negatively curved angle structures on admissible surfaces. This generalises Duncan and Howie's proof of the 1/2-lower bound in free groups, and can also be seen as a version of Bavard duality for letter-quasimorphisms.

math.GR↗

arXiv2023

Bavard duality for the relative Gromov seminorm

The relative Gromov seminorm is a finer invariant than stable commutator length where a relative homology class is fixed. We show a duality result between bounded cohomology and the relative Gromov seminorm, analogously to Bavard duality for scl. We give an application to computations of scl in graphs of groups. We also explain how our duality result can be given a purely algebraic interpretation via a relative version of the Hopf formula. Moreover, we show that this leads to a natural generalisation of a result of Calegari on a connection between scl and the rotation quasimorphism.

math.GT↗

arXiv2023

Isometric embeddings of surfaces for scl

Let $\varphi:F_1\to F_2$ be an injective morphism of free groups. If $\varphi$ is geometric (i.e. induced by an inclusion of oriented compact connected surfaces with nonempty boundary), then we show that $\varphi$ is an isometric embedding for stable commutator length. More generally, we show that if $T$ is a subsurface of an oriented compact (possibly closed) connected surface $S$, and $c$ is an integral $1$-chain on $\pi_1T$, then there is an isometric embedding $H_2(T,c)\to H_2(S,c)$ for the relative Gromov seminorm. Those statements are proved by finding an appropriate standard form for admissible surfaces and showing that, under the right homology vanishing conditions, such an admissible surface in $S$ for a chain in $T$ is in fact an admissible surface in $T$.

math.GT↗

arXiv2021

Free representations of outer automorphism groups of free products via characteristic abelian coverings

Given a free product $G$, we investigate the existence of faithful free representations of the outer automorphism group $\text{Out}(G)$, or in other words of embeddings of $\text{Out}(G)$ into $\text{Out}\left(F_m\right)$ for some $m$. This is based on a work of Bridson and Vogtmann in which they construct embeddings of $\text{Out}\left(F_n\right)$ into $\text{Out}\left(F_m\right)$ for some values of $n$ and $m$ by interpreting $\text{Out}\left(F_n\right)$ as the group of homotopy equivalences of a graph $X$ of genus $n$, and by lifting homotopy equivalences of $X$ to a characteristic abelian cover of genus $m$. Our construction for a free product $G$, using a presentation of $\text{Out}(G)$ due to Fuchs-Rabinovich, is written as an algebraic proof, but it is directly inspired by Bridson and Vogtmann's topological method and can be interpreted as lifting homotopy equivalences of a graph of groups. For instance, we obtain a faithful free representation of $\text{Out}(G)$ when $G=F_d\ast G_{d+1}\ast\cdots\ast G_n$, with $F_d$ free of rank $d$ and $G_i$ finite abelian of order coprime to $n-1$.

math.GR↗