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Mikolaj Fraczyk

Publications and source records attributed to Mikolaj Fraczyk.

At least 19 recordsLinked to original sources
arXiv2024

Minimal Submanifolds and Waists of Locally Symmetric Spaces

We study the higher expansion properties of locally symmetric spaces, with a particular focus on octonionic hyperbolic manifolds. We show that codimension two minimal submanifolds of compact octonionic locally symmetric spaces must have large volume, at least linear in the volume of the ambient space. As a corollary we prove linear waist inequalities for octonionic hyperbolic manifolds in codimension two and construct the first locally symmetric examples of power-law systolic freedom. We also show that any codimension two submanifold of small volume can be homotoped to a lower dimensional set. We use this to prove that branched covers of octonionic hyperbolic manifolds are stable in the sense of Dinur-Meshulam and to establish a uniform lower bound on the non-abelian Cheeger constants of octonionic hyperbolic manifolds. In a more general setting, we prove that maps from locally symmetric spaces to low dimensional euclidean spaces admit fibers whose fundamental group has large exponent of growth. We show as a consequence that cocompact lattices in $SL_n(\mathbb{R})$ have property $ FA_{\lfloor n/8\rfloor-1}$: any action on a contractible $CAT(0)$ simplicial complex of dimension at most $ \lfloor n/8\rfloor -1$ has a global fixed point.

math.DG↗
arXiv2024

Zariski dense non-tempered subgroups in higher rank of nearly optimal growth

We construct the first example of a Zariski-dense, discrete, non-lattice subgroup $\Gamma_0$ of a higher rank simple Lie group $G$, which is non-tempered in the sense that the quasi-regular representation $L^2(\Gamma_0\backslash G)$ is non-tempered. More precisely, let $n\ge 3$ and let $\Gamma$ be the fundamental group of a closed hyperbolic $n$-manifold that contains a properly embedded totally geodesic hyperplane. We show that there exists a non-empty open subset $\mathcal O$ of $\operatorname{Hom}(\Gamma, \operatorname{SO}(n,2))$ such that for any $\sigma\in \mathcal O$, the subgroup $\sigma(\Gamma)$ is a Zariski-dense and non-tempered Anosov subgroup of $\operatorname{SO}(n,2)$. In addition, the growth indicator of $\sigma(\Gamma)$ is nearly optimal: it almost realizes the supremum of growth indicators among all non-lattice discrete subgroups, a bound imposed by property $(T)$ of $\operatorname{SO}(n,2)$.

math.GR↗
arXiv2024

Automorphic density estimates and optimal Diophantine exponents

The present paper is devoted to establishing an optimal approximation exponent for the action of an irreducible uniform lattice subgroup of a product group on its proper factors. Previously optimal approximation exponents for lattice actions on homogeneous spaces were established under the assumption that the restriction of the automorphic representation to the stability group is suitably tempered. However, for irreducible lattices in semisimple algebraic groups, either this property does not hold or it amounts to an instance of the Ramanujan-Petersson-Selberg conjecture. Sarnak's Density Hypothesis and its variants bounding the multiplicities of irreducible representations occurring in the decomposition of the automorphic representation can be viewed as a weakening of the temperedness property. A refined form of this hypothesis has recently been established for uniform irreducible arithmetic congruence lattices arising from quaternion algebras. We employ this result in order to establish - unconditionally - an optimal approximation exponent for the actions of these lattices on the associated symmetric spaces. We also give a general spectral criterion for the optimality of the approximation exponent for irreducible uniform lattices in a product of arbitrary Gelfand pairs. Our methods involve utilizing the multiplicity bounds in the pre-trace formula, establishing refined estimates of the spherical transforms, and carrying out an elaborate spectral analysis that bounds the Hilbert-Schmidt norms of carefully balanced geometric convolution operators.

math.NT↗
arXiv2024

Co-spectral radius for countable equivalence relations

We define the co-spectral radius of inclusions $\mathcal{S}\leq \mathcal{R}$ of discrete, probability measure-preserving equivalence relations, as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $H\leq G$ of groups. For the proof, we develop a more general version of the 2-3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius.

math.PR↗
arXiv2024

Infinite Volume and Infinite Injectivity Radius

We prove the following conjecture of Margulis. Let $G$ be a higher rank simple Lie group and let $Λ\le G$ be a discrete subgroup of infinite covolume. Then, the locally symmetric space $Λ\backslash G/K$ admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups not necessarily associated to lattices. Yet, the result is new even for subgroups of infinite index of lattices. We establish similar results for higher rank semisimple groups with Kazhdan's property (T). We prove a stiffness result for discrete stationary random subgroups in higher rank semisimple groups and a stationary variant of the Stück-Zimmer theorem for higher rank semisimple groups with property (T). We also show that a stationary limit of a measure supported on discrete subgroups is almost surely discrete.

math.GR↗
arXiv2024

Spectral non-concentration near the top for unimodular random graphs

In recent work on equiangular lines, Jiang, Tidor, Yuan, Zhang, and Zhao showed that a connected bounded degree graph has sublinear second eigenvalue multiplicity. More generally they show that there cannot be too many eigenvalues near the top of the spectrum. We extend this result to infinite unimodular random graphs. As a corollary, the spectral distribution of the adjacency operator cannot have an atom at the top. For an infinite regular expander, we deduce that the singularity of the spectral measure at the top satisfies $μ_G[(1-θ)ρ,ρ] \lesssim θ^c$ for some constant $c>0$, where $ρ$ is the spectral radius of the adjacency operator of the graph. This implies new general estimates on the return probabilities of random walks.

math.PR↗
arXiv2023

Infinite volume and atoms at the bottom of the spectrum

Let $G$ be a higher rank simple real algebraic group, or more generally, any semisimple real algebraic group with no rank one factors and $X$ the associated Riemannian symmetric space. For any Zariski dense discrete subgroup $Γ<G$, we prove that $\operatorname{Vol}(Γ\backslash X)=\infty$ if and only if no positive Laplace eigenfunction belongs to $L^2(Γ\backslash X)$, or equivalently, the bottom of the $L^2$-spectrum is not an atom of the spectral measure of the negative Laplacian. This contrasts with the rank one situation where the square-integrability of the base eigenfunction is determined by the size of the critical exponent relative to the volume entropy of $X$.

math.SP↗
arXiv2023

Poisson-Voronoi tessellations and fixed price in higher rank

Let $G$ be a higher rank semisimple real Lie group or the product of at least two automorphism groups of regular trees. We prove all probability measure preserving actions of lattices in such groups have cost one, answering Gaboriau's fixed price question for this class of groups. We prove the minimal number generators of a torsion-free lattice in $G$ is sublinear in the co-volume of $Γ$, settling a conjecture of Abért-Gelander-Nikolov. As a consequence, we derive new estimates on the growth of first mod-$p$ homology groups of higher rank locally symmetric spaces. Our method of proof is novel, using low intensity Poisson point processes on higher rank symmetric spaces and the geometry of their associated Voronoi tessellations. We prove as the intensities limit to zero, these tessellations partition the space into ``horoball-like'' cells so that any two share an unbounded border. We use this new phenomenon to construct low cost graphings for orbit equivalence relations of higher rank lattices.

math.GT↗
arXiv2023

Discrete subgroups with finite Bowen-Margulis-Sullivan measure in higher rank

Let $G$ be a connected semisimple real algebraic group and $\Gamma<G$ be its Zariski dense discrete subgroup. We prove that if $\Gamma\backslash G$ admits any finite Bowen-Margulis-Sullivan measure, then $\Gamma$ is virtually a product of higher rank lattices and discrete subgroups of rank one factors of $G$. This may be viewed as a measure-theoretic analogue of classification of convex cocompact actions by Kleiner-Leeb and Quint, which was conjectured by Corlette in 1994. The key ingredients in our proof are the product structure of leafwise measures and the high entropy method of Einsiedler-Katok-Lindenstrauss. In a companion paper jointly with Edwards and Oh, we use this result to show that the bottom of the $L^2$ spectrum has no atom in any infinite volume quotient of a higher rank simple algebraic group.

math.DS↗
arXiv2023

Co-spectral radius of intersections

We study the behavior of the co-spectral radius of a subgroup $H$ of a discrete group $Γ$ under taking intersections. Our main result is that the co-spectral radius of an invariant random subgroup does not drop upon intersecting with a deterministic co-amenable subgroup. As an application, we find that the intersection of independent co-amenable invariant random subgroups is co-amenable.

math.GR↗
arXiv2022

On homology torsion growth

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups.

math.GT↗
arXiv2022

Bottom of the Length Spectrum of Arithmetic Orbifolds

We prove that cocompact arithmetic lattices in a simple Lie group are uniformly discrete if and only if the Salem numbers are uniformly bounded away from $1$. We also prove an analogous result for semisimple Lie groups. Finally, we shed some light on the structure of the bottom of the length spectrum of an arithmetic orbifold $Γ\backslash X$ by showing the existence of a positive constant $δ(X)>0$ such that squares of lengths of closed geodesics shorter than $δ$ must be pairwise linearly dependent over $\mathbb Q$.

math.GT↗
arXiv2022

On the optimal rate of equidistribution in number fields

Let $k$ be a number field. We study how well can finite sets of $\mathcal O_k$ equidistribute modulo powers of prime ideals, for all prime ideals at the same time. Our main result states that the optimal rate of equidistribution in $\mathcal O_k$ predicted by the local contstraints cannot be achieved unless $k=\mathcal Q$. We deduce that $\mathcal Q$ is the only number field where the ring of integers $\mathcal O_k$ admits a simultaneous $\frak p$-ordering, answering a question of Bhargava. Along the way we establish a non-trivial upper bound on the number of solutions $x\in \mathcal O_k$ of the inequality $|N_{k/\mathcal Q}(x(a-x))|\leq X^2$ where $X$ is a positive real parameter and $a\in\mathcal O_k$ is of norm at least $e^{-B}X$ for a fixed real number $B$. The latter can be translated as an upper bound on the average number of solutions of certain unit equations in $\mathcal O_k$.

math.NT↗
arXiv2020

Expander spanning subgraphs with large girth

We conjecture that finite graphs with positive Cheeger constant admit a spanning subgraph with positive Cheeger constant and girth proportional to the diameter. We prove this conjecture for regular expander graphs with large expansion. Our proof relies on the Local Lemma.

math.CO↗
arXiv2020

Strong Limit Multiplicity for arithmetic hyperbolic surfaces and $3$-manifolds

We show that every sequence of torsion-free arithmetic congruence lattices in $\mathrm{PGL}(2,\mathbb R)$ or $\mathrm{PGL}(2,\mathbb C)$ satisfies a strong quantitative version of the Limit Multiplicity property. We deduce that for $R>0$ in certain range, growing linearly in the degree of the invariant trace field, the volume of the $R$-thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic $3$-manifold $M$ is of order at most $\mathrm{Vol}(M)^{11/12}$. As an application we prove Gelander's conjecture on homotopy type of arithmetic hyperbolic $3$-manifolds: We show that there are constants $A,B$ such that every such manifold $M$ is homotopy equivalent to a simplicial complex with at most $A\mathrm{Vol}(M)$ vertices, all of degrees bounded by $B$.

math.NT↗
arXiv2019

Heat kernels are not uniform expanders

We study infinite analogues of expander graphs, namely graphs where subgraphs weighted by heat kernels form an expander family. Our main result is that there does not exist any infinite expander in this sense. This proves the analogue for random walks of Benjamini's conjecture that there is no infinite graph whose metric balls are uniformly expander. The proof relies on a study of stationary random graphs, in particular proving non-expansion of heat kernels in that setting. A key result is that any stationary random graph is stationary hyperfinite, which is potentially of independent interest.

math.CO↗
arXiv2018

Growth of mod$-2$ homology in higher rank locally symmetric spaces

Let $X$ be a higher rank symmetric space or a Bruhat-Tits building of dimension at least $2$ such that the isometry group of $X$ has property $(T)$. We prove that for every torsion free lattice $Γ\subset {\rm Isom} X$ any homology class in $H_1(Γ\backslash X,\mathbb F_2)$ has a representative cycle of total length $o_X({\rm Vol}(Γ\backslash X))$. As an application we show that $\dim_{\mathbb F_2} H_1(Γ\backslash X,\mathbb F_2)=o_X({\rm Vol}(Γ\backslash X)).$

math.AT↗