ON KAZHDAN CONSTANTS OF FINITE INDEX SUBGROUPS IN SLn(ℤ)

2012 ◽  
Vol 22 (03) ◽  
pp. 1250026
Author(s):  
UZY HADAD
Keyword(s):  
Finite Index ◽  
Congruence Subgroup ◽  
Infinite Family ◽  
Principal Congruence ◽  
The Other ◽  
Index Subgroup ◽  
Generating Set ◽  
Principal Congruence Subgroup ◽  
Finite Index Subgroup ◽  
Finite Index Subgroups

We prove that for any finite index subgroup Γ in SL n(ℤ), there exists k = k(n) ∈ ℕ, ϵ = ϵ(Γ) > 0, and an infinite family of finite index subgroups in Γ with a Kazhdan constant greater than ϵ with respect to a generating set of order k. On the other hand, we prove that for any finite index subgroup Γ of SL n(ℤ), and for any ϵ > 0 and k ∈ ℕ, there exists a finite index subgroup Γ′ ≤ Γ such that the Kazhdan constant of any finite index subgroup in Γ′ is less than ϵ, with respect to any generating set of order k. In addition, we prove that the Kazhdan constant of the principal congruence subgroup Γn(m), with respect to a generating set consisting of elementary matrices (and their conjugates), is greater than [Formula: see text], where c > 0 depends only on n. For a fixed n, this bound is asymptotically best possible.

scholarly journals Contractible, hyperbolic but non-CAT(0) complexes

2020 ◽  
Vol 30 (5) ◽  
pp. 1439-1463
Author(s):  
Richard C. H. Webb
Keyword(s):  
Isoperimetric Inequality ◽  
Finite Index ◽  
Mapping Class Group ◽  
Class Group ◽  
The Other ◽  
Mapping Class ◽  
Index Subgroup ◽  
Analogous Statement ◽  
Almost All ◽  
Finite Index Subgroup

AbstractWe prove that almost all arc complexes do not admit a CAT(0) metric with finitely many shapes, in particular any finite-index subgroup of the mapping class group does not preserve such a metric on the arc complex. We also show the analogous statement for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups. The obstruction is combinatorial. These complexes are all hyperbolic and contractible but despite this we show that they satisfy no combinatorial isoperimetric inequality: for any n there is a loop of length 4 that only bounds discs consisting of at least n triangles. On the other hand we show that the curve complexes satisfy a linear combinatorial isoperimetric inequality, which answers a question of Andrew Putman.

scholarly journals Congruence testing for odd subgroups of the modular group

2014 ◽  
Vol 17 (1) ◽  
pp. 206-208
Author(s):  
Thomas Hamilton ◽  
David Loeffler
Keyword(s):  
Finite Index ◽  
Modular Group ◽  
Congruence Subgroup ◽  
Index Subgroup ◽  
Finite Index Subgroup

AbstractWe give a computationally effective criterion for determining whether a finite-index subgroup of $\mathrm{SL}_2(\mathbf{Z})$ is a congruence subgroup, extending earlier work of Hsu for subgroups of $\mathrm{PSL}_2(\mathbf{Z})$.

scholarly journals Statistics of subgroups of the modular group

2021 ◽  
pp. 1-61
Author(s):  
Frédérique Bassino ◽  
Cyril Nicaud ◽  
Pascal Weil
Keyword(s):  
Finite Index ◽  
Modular Group ◽  
Large Deviation ◽  
Free Subgroup ◽  
Index Formula ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Finite Index Subgroup ◽  
Free Subgroups ◽  
Finite Index Subgroups

We count the finitely generated subgroups of the modular group [Formula: see text]. More precisely, each such subgroup [Formula: see text] can be represented by its Stallings graph [Formula: see text], we consider the number of vertices of [Formula: see text] to be the size of [Formula: see text] and we count the subgroups of size [Formula: see text]. Since an index [Formula: see text] subgroup has size [Formula: see text], our results generalize the known results on the enumeration of the finite index subgroups of [Formula: see text]. We give asymptotic equivalents for the number of finitely generated subgroups of [Formula: see text], as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size [Formula: see text] subgroup and prove a large deviation statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size [Formula: see text] subgroup (respectively, finite index subgroup, free subgroup) of [Formula: see text].

scholarly journals IRREDUCIBLE COXETER GROUPS

2007 ◽  
Vol 17 (03) ◽  
pp. 427-447 ◽  
Author(s):  
LUIS PARIS
Keyword(s):  
Direct Product ◽  
Finite Index ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Index Subgroup ◽  
Central Automorphism ◽  
Finite Coxeter Group ◽  
Irreducible Components ◽  
Finite Index Subgroup ◽  
Finite Index Subgroups

We prove that a non-spherical irreducible Coxeter group is (directly) indecomposable and that an indefinite irreducible Coxeter group is strongly indecomposable in the sense that all its finite index subgroups are (directly) indecomposable. Let W be a Coxeter group. Write W = WX1 × ⋯ × WXb × WZ3, where WX1, … , WXb are non-spherical irreducible Coxeter groups and WZ3 is a finite one. By a classical result, known as the Krull–Remak–Schmidt theorem, the group WZ3 has a decomposition WZ3 = H1 × ⋯ × Hq as a direct product of indecomposable groups, which is unique up to a central automorphism and a permutation of the factors. Now, W = WX1 × ⋯ × WXb × H1 × ⋯ × Hq is a decomposition of W as a direct product of indecomposable subgroups. We prove that such a decomposition is unique up to a central automorphism and a permutation of the factors. Write W = WX1 × ⋯ × WXa × WZ2 × WZ3, where WX1, … , WXa are indefinite irreducible Coxeter groups, WZ2 is an affine Coxeter group whose irreducible components are all infinite, and WZ3 is a finite Coxeter group. The group WZ2 contains a finite index subgroup R isomorphic to ℤd, where d = |Z2| - b + a and b - a is the number of irreducible components of WZ2. Choose d copies R1, … , Rd of ℤ such that R = R1 × ⋯ × Rd. Then G = WX1 × ⋯ × WXa × R1 × ⋯ × Rd is a virtual decomposition of W as a direct product of strongly indecomposable subgroups. We prove that such a virtual decomposition is unique up to commensurability and a permutation of the factors.

scholarly journals VIRTUAL ALGEBRAIC FIBRATIONS OF KÄHLER GROUPS

2019 ◽  
pp. 1-19
Author(s):  
STEFAN FRIEDL ◽  
STEFANO VIDUSSI
Keyword(s):  
Fundamental Group ◽  
Finite Index ◽  
Surface Groups ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Kähler Surfaces ◽  
Kähler Groups ◽  
Strong Relation ◽  
Finite Index Subgroup ◽  
Finite Index Subgroups

This paper stems from the observation (arising from work of Delzant) that “most” Kähler groups $G$ virtually algebraically fiber, that is, admit a finite index subgroup that maps onto $\mathbb{Z}$ with finitely generated kernel. For the remaining ones, the Albanese dimension of all finite index subgroups is at most one, that is, they have virtual Albanese dimension $va(G)\leqslant 1$ . We show that the existence of algebraic fibrations has implications in the study of coherence and higher BNSR invariants of the fundamental group of aspherical Kähler surfaces. The class of Kähler groups with $va(G)=1$ includes virtual surface groups. Further examples exist; nonetheless, they exhibit a strong relation with surface groups. In fact, we show that the Green–Lazarsfeld sets of groups with $va(G)=1$ (virtually) coincide with those of surface groups, and furthermore that the only virtually RFRS groups with $va(G)=1$ are virtually surface groups.

scholarly journals The dynamics of an Action of Sp(2n, Z)

2005 ◽  
Vol 71 (3) ◽  
pp. 399-404
Author(s):  
Anthony Nielsen
Keyword(s):  
Prime Number ◽  
Finite Index ◽  
Congruence Subgroup ◽  
Prime Number Theorem ◽  
Alternative Proof ◽  
Index Subgroup ◽  
Linear Action ◽  
Topologically Transitive ◽  
Finite Index Subgroup ◽  
Subgroup Theorem

S.G. Dani and S. Raghavan showed the linear action of Sp(2n,ℤ) on the space of symplectic p-frames for p ≤ n is topologically transitive. We give an alternative proof, from the prime number theorem and the congruence subgroup theorem, and show the action of every finite index subgroup of Sp(2n, ℤ) is topologically transitive.

scholarly journals Isomorphisms Between Big Mapping Class Groups

2018 ◽  
Vol 2020 (10) ◽  
pp. 3084-3099 ◽  
Author(s):  
Juliette Bavard ◽  
Spencer Dowdall ◽  
Kasra Rafi
Keyword(s):  
Finite Index ◽  
Outer Automorphism ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Index Subgroup ◽  
Class Groups ◽  
Infinite Type ◽  
Finite Index Subgroup ◽  
Finite Index Subgroups ◽  
Orientable Surfaces

Abstract We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these “big” mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.

scholarly journals Groups of p-deficiency one

2013 ◽  
Vol 156 (1) ◽  
pp. 115-121
Author(s):  
ANITHA THILLAISUNDARAM
Keyword(s):  
Finite Index ◽  
Index Subgroup ◽  
P Deficiency ◽  
Finitely Presented Groups ◽  
Finite Presentation ◽  
Finite Index Subgroup ◽  
Finitely Presented

AbstractIn a previous paper, Button and Thillaisundaram proved that all finitely presented groups of p-deficiency greater than one are p-large. Here we prove that groups with a finite presentation of p-deficiency one possess a finite index subgroup that surjects onto the integers. This implies that these groups do not have Kazhdan's property (T). Additionally, we show that the aforementioned result of Button and Thillaisundaram implies a result of Lackenby.

scholarly journals An Abramov formula for stationary spaces of discrete groups

2013 ◽  
Vol 34 (3) ◽  
pp. 837-853 ◽  
Author(s):  
YAIR HARTMAN ◽  
YURI LIMA ◽  
OMER TAMUZ
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Finite Index ◽  
Discrete Group ◽  
Return Time ◽  
Discrete Groups ◽  
Poisson Boundary ◽  
Index Subgroup ◽  
Expected Return ◽  
Finite Index Subgroup

AbstractLet $(G, \mu )$ be a discrete group equipped with a generating probability measure, and let $\Gamma $ be a finite index subgroup of $G$. A $\mu $-random walk on $G$, starting from the identity, returns to $\Gamma $ with probability one. Let $\theta $ be the hitting measure, or the distribution of the position in which the random walk first hits $\Gamma $. We prove that the Furstenberg entropy of a $(G, \mu )$-stationary space, with respect to the action of $(\Gamma , \theta )$, is equal to the Furstenberg entropy with respect to the action of $(G, \mu )$, times the index of $\Gamma $ in $G$. The index is shown to be equal to the expected return time to $\Gamma $. As a corollary, when applied to the Furstenberg–Poisson boundary of $(G, \mu )$, we prove that the random walk entropy of $(\Gamma , \theta )$ is equal to the random walk entropy of $(G, \mu )$, times the index of $\Gamma $ in $G$.

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