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Michael Brandenbursky

Publications and source records attributed to Michael Brandenbursky.

At least 19 recordsLinked to original sources
arXiv2026

Bounded cohomology of measure-preserving homeomorphism groups of non-orientable surfaces

Let $N_g$ be a closed non-orientable surface of genus $g\geq3$, and let $\operatorname{Homeo}_0(N_g,μ)$ be the identity component of its group of measure-preserving homeomorphisms. For every $n\geq2$ we construct a linear injection $\overline{EH}_b^n(F_2)\hookrightarrow\overline{EH}_b^n\bigl(\operatorname{Homeo}_0(N_g,μ)\bigr)$, where $\overline{EH}_b^n$ denotes reduced exact bounded cohomology. In particular, $H_b^2(\operatorname{Homeo}_0(N_g,μ))$ and $H_b^3(\operatorname{Homeo}_0(N_g,μ))$ are infinite-dimensional. The main new issue is low genus. A natural figure-eight subgroup $F_2\leqπ_1(N_g)$ is hyperbolically embedded for $g\geq4$, whereas in genus three hyperbolic embeddedness fails for every $π_1$-injective figure-eight with orientable regular neighborhood. We overcome this obstruction using the amalgam decomposition $π_1(N_g)=F_2*_{\mathbb Z}F_{g-2}$ and an isometric bounded-cohomology extension theorem for graphs of groups with amenable edge groups. Combined with the Gambaudo--Ghys transfer and annular pushing, this yields the injection above.

math.GT↗
arXiv2026

Non-commutative Barge-Ghys quasimorphisms: an erratum

This is an erratum to the paper ``Non-commutative Barge-Ghys quasimorphisms''. The effective version of Ambrose-Singer theorem used in this paper was mis-stated, making Theorem 3.11 false as stated. However, this statement remains true for a semidirect product $G=A\rtimes K$ of a non-compact Lie group with a compact Lie group $K$, as long as the curvature of the connection belongs to the Lie algebra of $K$. Theorems 3.12 and 3.22 remain true under this assumptions, the statements of Theorems 4.28 and 4.31 are modified accordingly. Theorem 5.2 is valid only for compact Lie groups.

math.DG↗
arXiv2026

The $L^p$-diameter of the space of contractible loops

We prove that the space of contractible simple loops of a given fixed area in any compact oriented surface has infinite diameter as a homogeneous space of the group of area-preserving diffeomorphisms endowed with the $L^p$-metric. As a special case, this resolves the $L^p$-metric analogue of the well-known question in symplectic topology regarding the space of equators on the two-sphere. Our methods involve a new class of functionals on a normed group, which are more general than quasi-morphisms.

math.GT↗
arXiv2026

On quasimorphisms and distortion in homeomorphism groups

Let $M$ be a smooth compact oriented connected manifold, and ${\rm Homeo}_0(M,μ)$ the group of homeomorphisms of $M$ supported away from $\partial M,$ which preserve a Borel probability measure $μ$ induced by a volume form on $M$, and are isotopic to the identity. In this paper, we identify those Gambaudo-Ghys and Polterovich quasimorphisms $Ψ\colon {\rm Diff}_0(M,μ)\to R$ which extend $C^0$-continuously to ${\rm Homeo}_0(M,μ)$ as quasimorphisms, and to ${\rm Homeo}_0(M)$ as group cochains whose differentials are semi-bounded cocycles. We present several applications of this result which include unboundedness of certain bi-invariant metric on the commutator subgroup of ${\rm Homeo}_0(M,μ)$, and conditions under which a homeomorphism in ${\rm Homeo}_0(M)$ is undistorted.

math.GT↗
arXiv2024

Volume and Euler classes in bounded cohomology of transformation groups

Let $M$ be an oriented smooth manifold, and $\operatorname{Homeo}(M,ω)$ the group of measure preserving homeomorphisms of $M$, where $ω$ is a finite measure induced by a volume form. In this paper we define volume and Euler classes in bounded cohomology of an infinite dimensional transformation group $\operatorname{Homeo}_0(M,ω)$ and $\operatorname{Homeo}(M,ω)$ respectively, and in several cases prove their non-triviality. More precisely, we define: - Volume classes in $\operatorname{H}_b^n(\operatorname{Homeo}_0(M,ω))$ where $M$ is a hyperbolic manifold of dimension $n$. - Euler classes in $\operatorname{H}_b^2(\operatorname{Homeo}(S,ω))$ where $S$ is a closed hyperbolic surface. We show that Euler classes have positive norms for any closed hyperbolic $S$ and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic $3$-manifolds, and hence they are non-trivial.

math.GT↗
arXiv2023

Cancelation norm and the geometry of biinvariant word metrics

We study biinvariant word metrics on groups. We provide an efficient algorithm for computing the biinvariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the biinvariant Cayley graph of a nonabelian free group. We investigate the geometry of cyclic subgroups. We observe that in many classes of groups cyclic subgroups are either bounded or detected by homogeneous quasimorphisms. We call this property the bq-dichotomy and we prove it for many classes of groups of geometric origin.

math.GT↗
arXiv2022

Lipschitz geometry of surface germs in $\mathbb{R}^4$: metric knots

A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in $\mathbb{R}^4$ is a topological knot (or link) in $S^3$. We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in $\mathbb{R}^4$ and the knot theory. Namely, for any knot $K$, we construct a surface $X_K$ in $\mathbb{R}^4$ such that: the link at the origin of $X_{K}$ is a trivial knot; the germs $X_K$ are outer bi-Lipschitz equivalent for all $K$; two germs $X_{K}$ and $X_{K'}$ are ambient bi-Lipschitz equivalent only if the knots $K$ and $K'$ are isotopic. We show that the Jones polynomial can be used to recognize ambient bi-Lipschitz non-equivalent surface germs in $\mathbb{R}^4$, even when they are topologically trivial and outer bi-Lipschitz equivalent.

math.AG↗
arXiv2021

Bounded cohomology of transformation groups

Let $M$ be a complete connected Riemannian manifold of finite volume. In this paper we present a new method of constructing classes in bounded cohomology of transformation groups such as $Homeo_0(M,μ)$, $Diff_0(M,vol)$ and $Symp_0(M,ω)$ (in case $M$ is symplectic). As an application we show that, under certain conditions on $π_1(M)$, the $3^{rd}$ bounded cohomology of these groups is infinite dimensional.

math.GT↗
arXiv2021

$C^0$-gap between entropy-zero Hamiltonians and autonomous diffeomorphisms of surfaces

Let $Σ$ be a surface equipped with an area form. There is an long standing open question by Katok, which, in particular, asks whether every entropy-zero Hamiltonian diffeomorphism of a surface lies in the $C^0$-closure of the set of integrable diffeomorphisms. A slightly weaker version of this question asks: ``Does every entropy-zero Hamiltonian diffeomorphism of a surface lie in the $C^0$-closure of the set of autonomous diffeomorphisms?'' In this paper we answer in negative the later question. In particular, we show that on a surface $Σ$ the set of autonomous Hamiltonian diffeomorphisms is not $C^0$-dense in the set of entropy-zero Hamiltonians. We explicitly construct examples of such Hamiltonians which cannot be approximated by autonomous diffeomorphisms.

math.DS↗
arXiv2021

The Schwarz-Milnor lemma for braids and area-preserving diffeomorphisms

We prove a number of new results on the large-scale geometry of the $L^p$-metrics on the group of area-preserving diffeomorphisms of each orientable surface. Our proofs use in a key way the Fulton-MacPherson type compactification of the configuration space of $n$ points on the surface due to Axelrod-Singer and Kontsevich. This allows us to apply the Schwarz-Milnor lemma to configuration spaces, a natural approach which we carry out successfully for the first time. As sample results, we prove that all right-angled Artin groups admit quasi-isometric embeddings into the group of area-preserving diffeomorphisms endowed with the $L^p$-metric, and that all Gambaudo-Ghys quasi-morphisms on this metric group coming from the braid group on $n$ strands are Lipschitz. This was conjectured to hold, yet proven only for low values of $n$ and the genus $g$ of the surface.

math.GT↗
arXiv2019

On the entropy norm on $Ham(S^2)$

In this note we prove that for each positive integer $m$ there exists a bi-Lipschitz embedding $Z^m\to Ham(S^2)$, where $Ham(S^2)$ is equipped with the entropy metric. In particular, the same result holds when the entropy metric is substituted with the autonomous metric.

math.GT↗
arXiv2019

Fragmentation norm and relative quasimorphisms

We prove that manifolds with complicated enough fundamental group admit measure-preserving homeomorphisms which have positive stable fragmentation norm with respect to balls of bounded measure.

math.GT↗
arXiv2019

Entropy and quasimorphisms

Let $S$ be a compact oriented surface. We construct homogeneous quasimorphisms on $Diff(S, area)$, on $Diff_0(S, area)$ and on $Ham(S)$ generalizing the constructions of Gambaudo-Ghys and Polterovich. We prove that there are infinitely many linearly independent homogeneous quasimorphisms on $Diff(S, area)$, on $Diff_0(S, area)$ and on $Ham(S)$ whose absolute values bound from below the topological entropy. In case when $S$ has a positive genus, the quasimorphisms we construct on $Ham(S)$ are $C^0$-continuous. We define a bi-invariant metric on these groups, called the entropy metric, and show that it is unbounded. In particular, we reprove the fact that the autonomous metric on $Ham(S)$ is unbounded.

math.GT↗
arXiv2018

Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups

Let $F_n$ be the free group on $n$ generators and $Γ_g$ the surface group of genus $g$. We consider two particular generating sets: the set of all primitive elements in $F_n$ and the set of all simple loops in $Γ_g$. We give a complete characterization of distorted and undistorted elements in the corresponding $Aut$-invariant word metrics. In particular, we reprove Stallings theorem and answer a question of Danny Calegari about the growth of simple loops. In addition, we construct infinitely many quasimorphisms on $F_2$ that are $Aut(F_2)$-invariant. This answers an open problem posed by Miklós Abért.

math.GT↗
arXiv2016

The $L^p$-diameter of the group of area-preserving diffeomorphisms of $S^2$

We show that for each $p \geq 1,$ the $L^p$-metric on the group of area-preserving diffeomorphisms of the two-sphere has infinite diameter. This solves the last open case of a conjecture of Shnirelman from 1985. Our methods extend to yield stronger results on the large-scale geometry of the corresponding metric space, completing an answer to a question of Kapovich from 2012. Our proof uses configuration spaces of points on the two-sphere, quasi-morphisms, optimally chosen braid diagrams, and, as a key element, the cross-ratio map $X_4(\mathbb{C} P^1) \to \mathcal{M}_{0,4} \cong \mathbb{C} P^1 \setminus \{\infty,0,1\}$ from the configuration space of $4$ points on $\mathbb{C} P^1$ to the moduli space of complex rational curves with $4$ marked points.

math.GT↗
arXiv2016

Concordance of certain 3-braids and Gauss diagrams

Let $β:=σ_1σ_2^{-1}$ be a braid in $B_3$, where $B_3$ is the braid group on 3 strings and $σ_1, σ_2$ are the standard Artin generators. We use Gauss diagram formulas to show that for each natural number $n$ not divisible by $3$ the knot which is represented by the closure of the braid $β^n$ is algebraically slice if and only if $n$ is odd. As a consequence, we deduce some properties of Lucas numbers.

math.GT↗