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A. Putman, A. Snowden
The Steinberg representation is irreducible
Duke Math. J. 172 (2023), no. 4, 775808.
Abstract:
We prove that the Steinberg representation of a connected reductive group over an infinite field is
irreducible. For finite fields, this is a classical theorem of Steinberg and Curtis.


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T. Brendle, N. Broaddus, A. Putman
The mapping class group of connect sums of S^{2} $\times$ S^{1}
Trans. Amer. Math. Soc. 376 (2023), 25572572.
Abstract:
Let M_{n} be the connect sum of n copies of S^{2} $\times$ S^{1}. A classical theorem of Laudenbach says
that the mapping class group Mod(M_{n})
is an extension of Out(F_{n}) by a group
($\mathbb{Z}$/2)^{n} generated
by sphere twists. We prove that this extension splits, so Mod(M_{n}) is the semidirect
product of Out(F_{n}) by ($\mathbb{Z}$/2)^{n}, which Out(F_{n}) acts on via the dual of the natural surjection
Out(F_{n}) $\rightarrow$ GL_{n}($\mathbb{Z}$/2). Our splitting takes
Out(F_{n}) to the subgroup of Mod(M_{n})
consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of M_{n}.
Our techniques also simplify various aspects of Laudenbach's original proof, including the identification
of the twist subgroup with ($\mathbb{Z}$/2)^{n}.


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A. Putman, S. Sam
VICmodules over noncommutative rings
Selecta Math. (N.S.) 28 (2022), no. 5, Paper No. 88.
Abstract:
For a finite ring R, not necessarily commutative, we prove that the category of VIC(R)modules over a left Noetherian ring k is locally Noetherian,
generalizing a theorem of the authors that dealt with commutative R. As an application, we
prove a very general twisted homology stability for GL_{n}(R) with R a finite noncommutative ring.


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A. Putman
The commutator subgroups of free groups and surface groups
Enseign. Math. 68 (2022), no. 34, 389408.
Abstract:
A beautifully simple free generating set for the commutator subgroup of a free group
was constructed by Tomaszewski. We give a new geometric proof of his theorem, and show
how to give a similar free generating set for the commutator subgroup of a surface group.
We also give a simple representationtheoretic description of the structure of the
abelianizations of these commutator subgroups and calculate their homology.


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T. Church, M. Ershov, A. Putman
On finite generation of the Johnson filtrations
J. Eur. Math. Soc. 24 (2022), no. 8, 28752914.
Abstract:
We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping
class group and the automorphism group of a free group is finitely generated in a linear stable range. This was originally
proved for the second terms by Ershov and He.


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A. Putman, D. Studenmund
The dualizing module and topdimensional cohomology group of GL_{n}($\mathcal{O}$)
Math. Z. 300 (2022), no. 1, 131.
Abstract:
For a number ring $\mathcal{O}$, Borel and Serre
proved that SL_{n}($\mathcal{O}$) is a virtual duality
group whose dualizing module is the Steinberg module. They also proved that
GL_{n}($\mathcal{O}$) is
a virtual duality group. In contrast to SL_{n}($\mathcal{O}$),
we prove that the dualizing
module of GL_{n}($\mathcal{O}$)
is sometimes the Steinberg module, but sometimes instead is
a variant that takes into account a sort of orientation.
Using this, we obtain vanishing and nonvanishing theorems for the cohomology of
GL_{n}($\mathcal{O}$) in its virtual cohomological dimension.


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J. Miller, P. Patzt, A. Putman
On the top dimensional cohomology groups of congruence subgroups of SL_{n}($\mathbb{Z}$)
Geom. Topol. 25 (2021), no. 2, 9991058.
Abstract:
Let $\Gamma$_{n}(p) be the levelp congruence subgroup of SL_{n}($\mathbb{Z}$). BorelSerre proved that
the cohomology of $\Gamma$_{n}(p) vanishes above degree $\binom{n}{2}$. We study the cohomology
in this top degree $\binom{n}{2}$. Let $\mathcal{T}$_{n}($\mathbb{Q}$) denote the Tits building of SL_{n}($\mathbb{Q}$).
LeeSzczarba conjectured that $H$^{$\binom{n}{2}$}($\Gamma$_{n}(p)) is isomorphic to
$\widetilde{H}$_{n2}($\mathcal{T}$_{n}($\mathbb{Q}$)/$\Gamma$_{n}(p)) and proved that this holds for p=3. We partially prove
and partially disprove this conjecture by showing that a natural map
$H$^{$\binom{n}{2}$}($\Gamma$_{n}(p)) $\rightarrow$ $\widetilde{H}$_{n2}($\mathcal{T}$_{n}($\mathbb{Q}$)/$\Gamma$_{n}(p))
is always surjective, but is only injective for p$\leq$5. In particular, we completely
calculate $H$^{$\binom{n}{2}$}($\Gamma$_{n}(5)) and improve known lower bounds for
the ranks of $H$^{$\binom{n}{2}$}($\Gamma$_{n}(p)) for p$\geq$5.


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D. Margalit, A. Putman
Surface groups, infinite generating sets, and stable commutator length
Proc. Roy. Soc. Edinburgh Sect. A. 150 (2020), no. 5, 23792386.
Abstract:
We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to
the generating set consisting of all simple closed curves.


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A. Putman, S. Sam, A. Snowden
Stability in the homology of unipotent groups
Algebra & Number Theory 14 (2020), no. 1, 119154.
Abstract:
Let R be a (not necessarily commutative) ring whose additive group is finitely generated and let
U_{n}(R) $\subset$ GL_{n}(R) be the group
of uppertriangular unipotent matrices over R. We study how the homology groups of
U_{n}(R) vary with n from the point of view of
representation stability. Our main theorem asserts that if for each n we have representations
M_{n} of U_{n}(R)
over a ring k that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule
[n] $\mapsto$ $\widetilde{H}$_{i}(U_{n}(R),M_{n})
defines a finitely generated OImodule. As a consequence, if k is a field then
dim $\widetilde{H}$_{i}(U_{n}(R),k)
is eventually equal to a polynomial in n. We also prove similar results for the Iwahori subgroups
of GL_{n}$\mathcal{O}$ for number rings $\mathcal{O}$.


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N. Fullarton, A. Putman
The highdimensional cohomology of the moduli space of curves with level structures
J. Eur. Math. Soc. 22 (2020), no. 4, 12611287.
Abstract:
We prove that the moduli space of curves with level structures has an enormous amount of rational cohomology
in its cohomological dimension. As an application, we prove that the coherent cohomological dimension of the
moduli space of curves is at least g2. Well known conjectures of Looijenga would imply that this is sharp.


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M. Kassabov, A. Putman
Equivariant group presentations and the second homology group of the Torelli group
Math. Ann. 376 (2020), no. 12, 227241.
Abstract:
We develop a theory of equivariant group presentations and relate them to the
second homology group of a group. Our main application says that
the second homology group of the Torelli subgroup of the mapping class group
is finitely generated as an Sp_{2g}($\mathbb{Z}$)module.


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J. Malestein, A. Putman
Simple closed curves, finite covers of surfaces, and power subgroups of Out(F_{n})
Duke Math. J. 168 (2019), no. 14, 27012726.
Abstract:
We construct examples of finite covers of punctured surfaces where the first rational
homology is not spanned by lifts of simple closed curves. More generally, for any
set $\mathcal{O} \subset$ F_{n} consisting of the union of finitely many Aut(F_{n})orbits, we construct
finiteindex normal subgroups of F_{n} whose first rational homology is not
spanned by powers of elements of $\mathcal{O}$.
These examples answer questions of FarbHensel, Looijenga, and Marché.
We also show that the normal
subgroup of Out(F_{n}) generated by k^{th} powers of transvections is often
infinite index. Finally, for any set $\mathcal{O} \subset$ F_{n} consisting of the union of finitely many Aut(F_{n})orbits,
we construct integral linear representations of free groups with
infinite image which maps all elements of $\mathcal{O}$ to torsion elements.


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T. Church, B. Farb, A. Putman
Integrality in the Steinberg module and the topdimensional cohomology of SL_{n}$\mathcal{O}$_{K}
Amer. J. Math. 141 (2019), no. 5, 13751419.
Abstract:
We prove a new structural result for the spherical Tits building attached to SL_{n} K for many number fields K, and more generally for the fraction fields of many Dedekind domains $\mathcal{O}$: the Steinberg module St_{n}(K) is generated by integral apartments if and only if the ideal class group cl($\mathcal{O}$) is trivial. We deduce this integrality by proving that the complex of partial bases of $\mathcal{O}$^{n} is CohenMacaulay.
We apply this to prove new vanishing and nonvanishing results for H^{$\nu$}(SL_{n}$\mathcal{O}$_{K};$\mathbb{Q}$), where $\mathcal{O}$_{K} is the ring of integers in a number field and $\nu$ is the virtual cohomological dimension of SL_{n}$\mathcal{O}$_{K}. The (non)vanishing depends on the (non)triviality of the class group of $\mathcal{O}$_{K}. We also obtain a vanishing theorem
for the cohomology H^{$\nu$}(SL_{n}$\mathcal{O}$_{K};V) with twisted coefficients V.


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A. Ash, A. Putman, S. Sam
Homological vanishing for the Steinberg representation
Compos. Math. 154 (2018), no. 6, 11111130.
Abstract:
For a field k, we prove that the i^{th} homology of the
groups GL_{n}(k),
SL_{n}(k),
Sp_{2n}(k),
SO_{n,n}(k),
and SO_{n,n+1}(k)
with coefficients in their Steinberg representations vanish for
n $\geq$ 2i+2.


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A. Putman
The Johnson homomorphism and its kernel
J. Reine Angew. Math. 735 (2018), 109141.
Abstract:
We give a new proof of a celebrated theorem of Dennis
Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup
of the mapping class group is generated by separating twists.
In fact, we prove a more general result that also applies to "subsurface Torelli groups". Using
this, we extend Johnson's calculation of the rational abelianization of the Torelli group
not only to the subsurface Torelli groups, but also to finiteindex subgroups of the Torelli
group that contain the kernel of the Johnson homomorphism.


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M. Day, A. Putman
On the second homology group of the Torelli subgroup of Aut(F_{n})
Geom. Topol. 21 (2017), no. 5, 28512896
Computer code used in paper here.
Abstract:
Let IA_{n} be the Torelli subgroup of Aut(F_{n}). We give an explicit
finite set of generators for H_{2}(IA_{n})
as a GL_{n}($\mathbb{Z}$)module. Corollaries
include a version of surjective representation stability for H_{2}(IA_{n}),
the vanishing of the GL_{n}($\mathbb{Z}$)coinvariants of H_{2}(IA_{n}), and the vanishing
of the second rational homology group of the level l congruence subgroup of
Aut(F_{n}). Our generating set is derived from a new group presentation for IA_{n}
which is infinite but which has a simple recursive form.


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T. Church, A. Putman
The codimensionone cohomology of SL_{n}$\mathbb{Z}$
Geom. Topol. 21 (2017), no. 2, 9991032.
Abstract:
We prove that H^{$\binom{n}{2}$1}(SL_{n} $\mathbb{Z}$;$\mathbb{Q}$) = 0,
where $\binom{n}{2}$ is the cohomological dimension
of SL_{n}$\mathbb{Z}$, and similarly for GL_{n}$\mathbb{Z}$. We also prove analogous vanishing theorems
for cohomology with coefficients in a rational representation
of the algebraic group GL_{n}.
These theorems are derived from a presentation of the Steinberg
module for SL_{n}$\mathbb{Z}$
whose generators are integral apartment classes, generalizing Manin's presentation for the Steinberg module of
SL_{2}$\mathbb{Z}$. This presentation was originally constructed by Bykovskii. We give a new
topological proof of it.


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A. Putman, S. Sam
Representation stability and finite linear groups
Duke Math. J. 166 (2017), no. 13, 25212598.
Abstract:
We construct analogues of FImodules where the role of the symmetric group
is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity.
Applications include a proof of Schwartz's Artinian conjecture in the generic representation theory of finite fields and
representationtheoretic versions of homological stability for congruence subgroups
of the general linear group, the automorphism group of a free group, the symplectic
group, and the mapping class group.


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M. Day, A. Putman
A Birman exact sequence for the Torelli subgroup of Aut(F_{n})
Internat. J. Algebra Comput. 26 (2016), no. 3, 585617.
Computer code used in paper here.
Abstract:
We develop an analogue of the Birman exact sequence for the Torelli subgroup of
Aut(F_{n}).
This builds on earlier work of the authors who studied an analogue of the
Birman exact sequence for the entire group
Aut(F_{n}). These results play
an important role in the authors' recent work on the second homology group of the Torelli group.


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J. Malestein, A. Putman
PseudoAnosov dilatations and the Johnson filtration
Groups Geom. Dyn. 10 (2016), no. 2, 771793.
Abstract:
Answering a question of FarbLeiningerMargalit, we give explicit lower bounds for the dilatations of pseudoAnosov mapping classes lying in the d^{th} term of the Johnson filtration of the mapping class group.


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A. Putman
Stability in the homology of congruence subgroups
Invent. Math. 202 (2015), no. 3, 9871027.
Abstract:
The homology groups of many natural sequences of groups $\{G_n\}_{n=1}^{\infty}$
(e.g. general linear groups, mapping class groups, etc.)
stabilize as $n \rightarrow \infty$. Indeed, there is a wellknown machine
for proving such results that goes back to early work of Quillen. Church
and Farb discovered that many sequences of groups whose homology groups
do not stabilize in the classical sense actually stabilize in some sense
as representations. They called this phenomena {\em representation stability}.
We prove that the homology groups of congruence subgroups of GL_{n}(R) (for
almost any reasonable ring R) satisfy a strong version of representation stability that we call
central stability. The definition of central stability is very different
from ChurchFarb's definition of representation stability (it is defined via
a universal property), but we prove that it implies representation stability.
Our main tool is a new machine for proving central stability that is analogous
to the classical homological stability machine.


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T. Church, A. Putman
Generating the Johnson filtration
Geom. Topol. 19 (2015), no. 4, 2217–2255.
Abstract:
For k$\geq$1, let $\mathcal{I}_g^1$(k) be the k^{th} term in the Johnson filtration
of the mapping class group of a genus g surface with one boundary component.
We prove that for all k$\geq$1, there exists some G_{k} such that $\mathcal{I}_g^1$(k) is generated
by elements which are supported on subsurfaces whose genus is at most G_{k}. We also prove
similar theorems for the Johnson filtration of Aut(F_{n}) and
for certain modp analogues of the Johnson filtrations of both the mapping class group and
of Aut(F_{n}). The main tools used in the proofs are the related theories of FImodules (due to the
first author together with Ellenberg and Farb) and central stability (due to the second author), both
of which concern the representation theory of the symmetric groups over $\mathbb{Z}$.


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T. Brendle, D. Margalit, A. Putman
Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=1
Invent. Math. 200 (2015), no. 1, 263310.
Abstract:
We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=1 and also the fundamental group of the branch locus of the period mapping. One application is that each component in Torelli space of the
locus of hyperelliptic curves becomes simplyconnected when curves of compact type are added.


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T. Church, B. Farb, A. Putman
A stability conjecture for the unstable cohomology of SL_{n}($\mathbb{Z}$), mapping class groups, and Aut(F_{n})
in "Algebraic Topology: Applications and New Directions", 5570,
Contemp. Math. 620 (2014), Amer. Math. Soc., Providence, RI.
Abstract:
In this paper we give a conjectural picture of a large piece of the unstable rational cohomology
of SL_{n}($\mathbb{Z}$), of mapping class groups, and of Aut(F_{n}).


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A. Putman, B. Wieland
Abelian quotients of subgroups of the mapping class group and higher Prym representations
J. London Math. Soc. 88 (2013), no. 1, 7996.
Abstract:
A wellknown conjecture asserts that the mapping class group
of a surface (possibly with punctures/boundary) does not
virtually surject onto $\mathbb{Z}$ if the genus of the surface is large.
We prove that if this conjecture holds for some genus, then it
also holds for all larger genera. We also prove that if there
is a counterexample to this conjecture, then there must
be a counterexample of a particularly simple form. We prove
these results by relating the conjecture to a family of
linear representations of the mapping class group that we
call the higher Prym representations. They generalize
the classical symplectic representation.


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M. Day, A. Putman
The complex of partial bases for F_{n} and finite generation of the Torelli subgroup of Aut(F_{n})
Geom. Dedicata 164 (2013), 139153.
Abstract:
We study the complex of partial bases of a free group, which is
an analogue for Aut(F_{n}) of the curve complex for the mapping class
group. We prove that it is connected and simply connected, and we
also prove that its quotient by the Torelli subgroup of Aut(F_{n})
is highly connected. Using these results, we give a new, topological
proof of a theorem of Magnus that asserts that the Torelli subgroup
of Aut(F_{n}) is finitely generated.


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A. Putman
The Torelli group and congruence subgroups of the mapping class group
in "Moduli spaces of Riemann surfaces (Park City, UT, 2011)", 167194,
IAS/Park City Math. Ser. 20 (2013), Amer. Math. Soc., Providence, RI.
Abstract:
These are the lecture notes for my course at the 2011 Park City Mathematics
Graduate Summer School. The first two lectures covered the basics of the Torelli
group and the Johnson homomorphism, and the third and fourth lectures
discussed the second cohomology group of the level p congruence subgroup
of the mapping class group, following my papers "The second rational
homology group of the moduli space of curves with level structures" and
"The Picard group of the moduli space of curves with level structures".


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T. Church, B. Farb, A. Putman
The rational cohomology of the mapping class group vanishes in its virtual cohomological dimension
Int. Math. Res. Not. (2012), no. 21, 5025–5030.
Abstract:
Let Mod_{g} be the mapping class group of a genus g$\geq$2 surface. The
group Mod_{g} has virtual cohomological dimension 4g5. In this note we use a theorem of
Broaddus and the combinatorics of chord diagrams to prove that
H^{4g5}(Mod_{g};$\mathbb{Q}$)=0.


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A. Putman
Small generating sets for the Torelli group
Geom. Topol. 16 (2012), no. 1, 111–125.
Abstract:
Proving a conjecture of Dennis Johnson, we show
that the Torelli subgroup of the mapping class group has a finite generating set
whose size grows cubically with respect to the genus of the surface. Our main tool
is a new space (the handle graph of a surface) on which the Torelli group acts cocompactly.


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M. Day, A. Putman
A Birman exact sequence for Aut(F_{n})
Adv. Math. 231 (2012), 243–275.
Abstract:
The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of
gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism
group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid group. We prove
that in the context of the automorphism group of a free group, the natural kernel is finitely generated.
However, it is not finitely presentable; indeed, we prove that its second rational homology group has
infinite rank by constructing an explicit infinite collection of linearly independent abelian cycles.
We also determine the abelianization of our kernel and build a simple infinite presentation for it.
The key to many of our proofs are several new generalizations of the Johnson homomorphisms.


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A. Putman
The Picard group of the moduli space of curves with level structures
Duke Math. J. 161 (2012), no. 4, 623–674.
Computer code used in paper here.
Abstract:
For 4$\nmid$L and g large, we calculate the integral Picard
groups of the moduli spaces of curves and principally polarized abelian varieties with level L
structures. In particular, we determine the divisibility properties of the standard line
bundles over these moduli spaces and we calculate the second integral cohomology group of the level L subgroup
of the mapping class group (in a previous paper, the author determined this rationally).
This entails calculating the abelianization of the level L subgroup
of the mapping class group, generalizing previous results of Perron, Sato, and the author. Finally,
along the way we calculate the first homology group of the mod L symplectic group with
coefficients in the adjoint representation.


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A. Putman
The second rational homology group of the moduli space of curves with level structures
Adv. Math. 229 (2012), 1205–1234.
Abstract:
Let $\Gamma$ be a finiteindex subgroup of the mapping
class group of a closed genus g surface that contains the Torelli group. For
instance, $\Gamma$ can be the level L subgroup or the spin mapping
class group. We show that H_{2}($\Gamma$;$\mathbb{Q}$) $\cong$ $\mathbb{Q}$
for g$\geq$5. A
corollary of this is that the rational Picard groups of the associated
finite covers of the moduli space of curves are equal to $\mathbb{Q}$. We also prove
analogous results for surface with punctures and boundary components.


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A. Putman
Abelian covers of surfaces and the homology of the level L mapping class group
J. Topol. Anal. 3 (2011), no. 3, 265–306.
Abstract:
We calculate the first homology group of the mapping class group with coefficients in the first rational homology group
of the universal abelian $\mathbb{Z}$/L$\mathbb{Z}$cover of the surface. If the surface has one marked point, then the answer
is $\mathbb{Q}$^{$\tau$(L)}, where $\tau$(L) is the number of positive divisors of L. If the surface instead has
one boundary component, then the answer is $\mathbb{Q}$. We also perform the same calculation for the level L subgroup
of the mapping class group. Set
H_{L}=H_{1}($\Sigma$_{g};$\mathbb{Z}$/L$\mathbb{Z}$).
If the surface has one marked point, then the answer is $\mathbb{Q}$[H_{L}], the rational group ring of H_{L}.
If the surface instead has one boundary component, then the answer is $\mathbb{Q}$.


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A. Putman
Obtaining presentations from group actions without making choices
Algebr. Geom. Topol. 11 (2011), 17371766.
Abstract:
Consider a group G acting nicely on a simplyconnected
simplicial complex X. Numerous classical methods exist
for using this group action to produce a presentation for G. For
the case that X/G is 2connected, we give a new method that
has the novelty that one does not have to identify a fundamental domain
for the action. Indeed, the resulting presentation is canonical in
the sense that no arbitrary choices need to be made. It can be viewed
as a nonabelian analogue of a simple result in the study of equivariant homology.


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N. Broaddus, B. Farb, A. Putman
Irreducible Sprepresentations and subgroup distortion in the mapping class group
Comment. Math. Helv. 86 (2011), 537556.
Abstract:
We prove that various subgroups of the mapping
class group Mod($\Sigma$) of a surface $\Sigma$
are at least exponentially distorted. Examples include the
Torelli group (answering a question of Hamenstädt), the
"pointpushing" and surface braid subgroups,
and the Lagrangian subgroup. Our techniques include
a method to compute lower bounds on distortion via
representation theory and an extension of Johnson theory to arbitrary
subgroups of H_{1}($\Sigma$;$\mathbb{Z}$).


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J. Malestein, A. Putman
On the selfintersections of curves deep in the lower central series of a surface group
Geom. Dedicata 149 (2010), no. 1, 73–84.
Abstract:
We give various estimates of the minimal number of selfintersections of a nontrivial
element of the k^{th} term of the lower central series and derived series of the fundamental
group of a surface. As an application, we obtain a new topological proof of the fact that
free groups and fundamental groups of closed surfaces are residually nilpotent. Along the
way, we prove that a nontrivial element of the k^{th} term of the lower central series
of a nonabelian free group has to have word length at least k in a free generating set.


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A. Putman
A note on the abelianizations of finiteindex subgroups of the mapping class group
Proc. Amer. Math. Soc. 138 (2010), 753758.
Abstract:
For some g$\geq$3, let $\Gamma$ be a finite index subgroup of the mapping class group
of a genus g surface (possibly with boundary components and punctures).
An old conjecture of Ivanov says that the abelianization of $\Gamma$ should be finite.
In this note, we prove two theorems supporting this conjecture. For the first,
let $T_x$ denote the Dehn twist about a simple closed curve x. For some n$\geq$1,
we have $T_x^n \in \Gamma$. We prove that $T_x^n$ is torsion in the abelianization
of $\Gamma$. Our second result shows that the abelianization of $\Gamma$ is finite
if $\Gamma$ contains a "large chunk" (in a certain technical sense) of the
Johnson kernel, that is, the subgroup of the mapping class group generated by
twists about separating curves.


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A. Putman
An infinite presentation of the Torelli group
Geom. Funct. Anal. 19 (2009), no. 2, 591643.
Abstract:
In this paper, we construct a presentation of the Torelli subgroup of the mapping
class group of a surface whose generators consist of the set of
all "separating twists", all "bounding pair maps", and
all "commutators of simply intersecting pairs" and whose relations all come from
a short list of topological configurations of these generators on the surface. Aside from a few obvious ones, all of
these relations come from a set of embeddings of groups derived from surface groups into
the Torelli group. In the process of analyzing these embeddings, we derive a novel
presentation for the fundamental group of a closed surface whose generating set is the
set of all simple closed curves.
Our main tool for analyzing the Torelli group is a new theorem which
allows us to obtain presentations for groups
acting on simplicial complexes without identifying a fundamental domain. We apply this
to the action of the Torelli group on a variant of the complex of curves, yielding an
inductive description of the Torelli group in terms of the the subgroups stabilizing
simple closed curves on the surface.


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J. Birman, D. Johnson, A. Putman
Symplectic Heegaard splittings and linked abelian groups
in "Groups of Diffeomorphisms", 135220,
Adv. Stud. Pure Math. 52 (2008), Math. Soc. Japan, Tokyo.
Abstract:
Let f be the gluing map of a Heegaard splitting of a 3manifold W. The
goal of this paper is to determine the information about W contained in the
image of f under the symplectic representation of the mapping class group. We
prove three main results. First, we show that the first homology group of the
three manifold together with Seifert's linking form provides a complete set of
stable invariants. Second, we give a complete, computable set of invariants for
these linking forms. Third, we show that a slight augmentation of Birman's
determinantal invariant for a Heegaard splitting gives a complete set of
unstable invariants.


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A. Putman
A note on the connectivity of certain complexes associated to surfaces
Enseign. Math. 54 (2008), no. 2, 287301.
Abstract:
This note is devoted to a trick which yields almost trivial
proofs that certain complexes associated to topological surfaces are connected
or simply connected. Applications include new proofs that the
complexes of curves, separating curves, nonseparating curves, pants,
and cut systems are all connected for genus g$\gg$0. We also prove that
two new complexes are connected : one involves curves which split
a genus 2g surface into two genus g pieces, and the other
involves curves which are homologous to a fixed curve. The connectivity
of the latter complex can be interpreted as saying the "homology"
relation on the surface is (for g$\geq$3) generated
by "embedded/disjoint homologies". We finally prove that the
complex of separating curves is simply connected for g$\geq$4.


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N. Broaddus, B. Farb, A. Putman
The Casson invariant and the word metric on the Torelli group
C. R. Acad. Sci. Paris, Ser. I 345 (2007), 449452.
Abstract:
We bound the value of the Casson invariant of any integral homology 3sphere M by
a constant times the distancesquared to the identity, measured in any word metric on
the Torelli group $\mathcal{I}$, of the element of $\mathcal{I}$ associated to any Heegaard splitting of M. We
construct examples which show this bound is asymptotically sharp.


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A. Putman
Cutting and pasting in the Torelli group
Geom. Topol. 11 (2007), no. 2, 829865.
Abstract:
We introduce machinery to allow "cutandpaste"style inductive
arguments in the Torelli subgroups of the mapping class groups. In
the past these arguments have been problematic because the restriction
of the Torelli groups to subsurfaces gives different groups depending
on how the subsurfaces are embedded. We define a category TSur whose
objects are surfaces together with a decoration restricting how they can be embedded
into larger surfaces and whose morphisms are embeddings which respect the
decoration. There is a natural "Torelli functor" on this category which
extends the usual definition of the Torelli group on a closed surface. Additionally,
we prove an analogue of the Birman exact sequence for the Torelli groups of
surfaces with boundary and use the action of the Torelli groups on the complex
of curves to find generators for the Torelli groups. For genus g$\geq$1 only
twists about (certain) separating curves and bounding pairs are needed, while
for genus g=0 a new type of generator (a "commutator of a simply intersecting
pair") is needed. As a special case, our methods provide a new,
more conceptual proof of the classical result
of BirmanPowell which says that the Torelli groups on closed surfaces
are generated by twists about separating curves and bounding pairs.


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A. Putman
The rationality of sol manifolds
J. Algebra 304 (2006), no. 1, 190215.
Abstract:
Let $\Gamma$ be the fundamental group of a manifold modeled on three dimensional
Sol geometry. We prove that $\Gamma$ has a finite index subgroup G which
has a rational growth series with respect to a natural generating set. We do
this by enumerating G by a regular language. However, in contrast to most
earlier proofs of this sort our regular language is not a language of words
in the generating set, but rather reflects a different geometric structure
in G.
